Beginning of Section AbstrHF
Variable ain asubq adisjoint : set → set → prop
Variable a0 a1 a2 a3 a4 : set
Variable apow asing : set → set
Variable aun aint asm : set → set → set
Variable aal2 aal3 aal4 aal5 aal6 aal7 aex2 aex3 aex4 aex5 aex6 : set → prop
Theorem. (conj_AbstrHF_TMPjX7hFaPjbcqznm5PuccMDMpyvDBCSQ2z)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → ain a1 a3 → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a3 a2) → (¬ (a0 = a2)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a2) a1) → (¬ ain (asing a1) (asing a2)) → asubq a1 (asm a3 (asing a1)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ ain (apow a2) (asing a2)) → (¬ (a2 = asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing a2) a3) → (¬ ain (apow a2) a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → asubq a3 (apow a2) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → ain a3 (asm a4 (asing a1)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (asm a3 (asing a2) = a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a4 (asing a3)) a3 → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) a4 → asubq (asm (apow a2) (apow (asing a1))) a4 → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing a3)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (asm a4 (asing a2)) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm a3 (asing a1)) → aex2 (aun a1 (asing a3)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex3 (asm a4 (asing a3)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2)
In Proofgold the corresponding term root is 931870... and proposition id is 04596e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMEswXexHDuzYejc2UdX6zDb8tDKopGLZWh)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a2 a1) → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ ain (asing a1) a0) → (¬ (a0 = asing a1)) → (¬ ain a1 (asing a2)) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing (asing a1)) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) a4 → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 a3) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 (aun (asing a1) (asing (asing a1))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → ain a1 (asm a4 (apow (asing a2)))
In Proofgold the corresponding term root is e3edb9... and proposition id is f376fe...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMV1dJsWuxDRQx7yhnNq4shrepxnCm5Mfqw)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a0 a3 → ain a2 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = asm a3 (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (asing a2) a3) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a3 (asing a3) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm a4 (asing a3)) a3 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (apow a2)) → aal4 (aun a3 (asing (asing a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (asing (asing a1)) a4) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 a3 → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 a4 → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → aex5 (aun (asing (asing a1)) a4)
In Proofgold the corresponding term root is 3386a4... and proposition id is 1eecb5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUjENah9uQtGHEC8SehurKemd3pcXDxNmo)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (¬ asubq a4 a3) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a1))) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ (a2 = asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain a2 (aun a1 (asing a3))) → ain a3 (aun (asing a1) (asing a3)) → ain a0 (asm a4 (asing a2)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asing a2) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain a1 (aun a2 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asing a1) (asm a2 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a4 (asing a3)) a3 → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 a3) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 a3 → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))
In Proofgold the corresponding term root is 9a0600... and proposition id is 33fd64...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQ8YMpcc6otJGo5ig6eZztM1raUxB1VZd7)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain a1 (asing a1) → (¬ ain a1 (apow (asing a1))) → ain a2 (asm (apow a2) a2) → (¬ (a1 = asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ asubq (asing a2) a2) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (asing (asing a1) = a3)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a3 (asing a3) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (asing a2)) → ain a3 (asm a4 (apow (asing a2))) → ain a1 (aun (asing (asing a2)) a4) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal3 (asm (apow a2) (asing a2)) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal5 (aun (asing (apow (asing a1))) a4) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3)))
In Proofgold the corresponding term root is 2aaf99... and proposition id is e2a612...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMasiMMQCzjU6xnuoQytrmuAsNGmTduLQdo)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (asing a1) → ain a2 (asing a2) → (¬ (a0 = asing a2)) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain a1 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ (a2 = asing a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ ain (apow a2) a3) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain a1 (apow (apow (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a1 (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (apow a2) (asing (asing a1))) a3 → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (asm (apow a2) (apow (asing a1))) → aal5 (aun (asing (asing a2)) a4) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aun a1 (asing (apow a2))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a1 (asing (apow a2)))
In Proofgold the corresponding term root is 5f17c1... and proposition id is 026fd7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSrRU1qadDKJRBTEpzwu1TMt72rRNWUth4)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (¬ (a0 = a1)) → ain a0 (apow (asing a1)) → ain a0 (apow a2) → (¬ ain a0 (asing (asing a1))) → ain (asing a1) (apow (asing a1)) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ asubq (apow a2) (asing a2)) → (¬ (a1 = apow a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (apow (asing a1)) a4) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal2 (asm (apow a2) (apow (asing a1))) → aex2 (apow (asing a1)) → aex2 (asm a4 a2) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3)))
In Proofgold the corresponding term root is 15159d... and proposition id is e2b2d3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYrmjMfi2wkPTiPMV9oVBPzaEW71AJPQQN)
(∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a0) → (¬ ain a3 a3) → (¬ (a1 = a2)) → (¬ (a2 = a3)) → (¬ ain a0 (asing a2)) → (¬ (a0 = asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ asubq a1 (asing a2)) → ain a0 (apow (asing a2)) → (¬ ain (apow a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a3 (asm a4 (apow (asing a2))) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain a0 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm a4 (asing a3) = a3) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing a2)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (apow a2) (apow (asing a1))) → aex3 (aun (asing a2) (apow (asing a2))) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)))
In Proofgold the corresponding term root is 5cba9a... and proposition id is 2d0ede...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWK3whEP9K6GZdcVcRVnP11ayzfumzVy6X)
(∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → ain a0 (asm a3 (asing a1)) → ain a1 (asm (apow a2) (asing a2)) → asubq a1 (apow (asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain a2 (apow (asing a2))) → (¬ (a1 = asing a2)) → (¬ (a1 = asm a3 (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ ain (asing a2) a2) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun a3 (asing (asing a2))) → ain a0 (aun a1 (asing a3)) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a1 (aun a3 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aun (asing a1) (asing (asing a1))) → aex3 a3 → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ (a1 = asing (asing a1)))
In Proofgold the corresponding term root is a1d84f... and proposition id is cdad56...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcvcUgXgZWH7st62SXET8yoNUPPdjjA9Vb)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a0 a1 → ain a0 a2 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (¬ ain a3 a2) → (¬ ain (asing a1) a0) → (¬ asubq a1 (asing a2)) → (¬ ain (asing a1) a1) → (¬ ain (apow a2) a2) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → asubq a3 (apow a2) → (¬ (asing a2 = asm (apow a2) a2)) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing (asing a1)) a4) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (asing (apow a2)) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (asm a4 (asing a2)) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal3 (asm (apow a2) (asing a1)) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 a3 → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (asm a4 (asing a3)) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is 1c43f9... and proposition id is eba22c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRePohUuhqWSrbHfgd9WRjSapn4erpgJfp)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a3 a3) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (¬ ain (asing a1) a0) → (¬ ain a0 (asing a1)) → ain a1 (apow a2) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain a0 (asm a4 a2)) → (¬ ain a3 (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq a1 (asm a2 (asing a1)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal2 (apow (asing (asing a1))) → aex3 (asm (apow a2) (asing a2)) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (asm a4 (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex4 (aun a3 (asing (apow a2))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ (asm a3 (asing a1) = a3))
In Proofgold the corresponding term root is 204ee3... and proposition id is a161fb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMaZ6izwx7ynyoJtRPMYKMo2NuBaKp18mnj)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → asubq a1 a2 → ain a1 (apow a2) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → ain a2 (apow a2) → (¬ ain a3 (asing a1)) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ (apow (asing a1) = a3)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain a0 (apow (asing (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a0 (aun a1 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (aun a2 (asing (apow a2))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aun a1 (asing a3)) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a3 (asing (apow a2)))
In Proofgold the corresponding term root is a8d818... and proposition id is 6c1ee0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLXMjxw1gHcyW8opMTFmF7kza4jZeQzikA)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (¬ ain a1 a1) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a0 (apow a2) → (¬ (a0 = asing (asing a1))) → ain a1 (asm (apow a2) (asing a2)) → ain a2 (asm a3 (asing a1)) → (¬ ain (asing a1) (asing a1)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (asing (asing a1))) → (¬ (a2 = asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = apow a2)) → ain a0 (apow (apow (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal4 (aun a3 (asing (asing a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (apow (asing a2)) → aex3 (aun a2 (asing (apow a2))) → aex3 (asm a4 (asing a0)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asing (asing a2)) a4) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))
In Proofgold the corresponding term root is 47bfee... and proposition id is 2b2d49...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVV4ZVBFaRbJvhVkhnAh6ueGz5BeqDTpHx)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a1 a3 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (¬ ain a2 a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (asing a1 = asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a3 (asing a3) → ain a3 (asm a4 (asing a2)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal5 (aun (apow a2) (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (apow (asing a1)) → aex2 (asm a4 a2) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → aex5 (aun (apow a2) (asing (apow a2)))
In Proofgold the corresponding term root is 281644... and proposition id is 5852a6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFtZJmNXqu1edLTpT3djhn4q9VtMvannRo)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ (asing a1 = a2)) → (¬ asubq a2 (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) a3) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain (apow (asing a1)) a4) → (¬ ain (apow a2) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → (¬ ain a0 (asm a4 a2)) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq a3 (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal3 a3 → aal3 (asm (apow a2) (asing a1)) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aun a1 (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun a3 (asing (apow (asing a1)))) → aex3 (asm a4 (asing a0)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (aun (asing (apow (asing a1))) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → ain (apow a2) (asing (apow a2))
In Proofgold the corresponding term root is 5a3322... and proposition id is 4a37c8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TML1gK82iBRzrppPbhtHPeA2jXRQpY8JdhN)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → ain a0 (asm (apow a2) (asing a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain (apow (asing a1)) a4) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a1 (aun (asing (asing a2)) a4) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 a1) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 a4) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (asm a4 (asing a3)) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1)))
In Proofgold the corresponding term root is aa0d00... and proposition id is 4dc78a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFLhP5tYUmmzDbv98DqREaNPLaDBZzmH1L)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, asubq X24 X24) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a2 a1) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (¬ ain a0 (asing a1)) → (¬ (asing a1 = a2)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq a3 (apow a2) → (¬ (a3 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asing a2 = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a1 (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm a2 (asing a1)) a1 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal3 (apow (asing a1))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → aal3 (asm a4 (asing a1)) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (apow a2) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, (¬ aal3 (apow (asing X24))))
In Proofgold the corresponding term root is 19b5a1... and proposition id is 394664...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWaHeCVmk1eyaLwfCyrYNV1kERS5dvA9xa)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a0 (asm a3 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → ain a2 (apow a2) → (¬ (a1 = asing a2)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (asm (apow a2) a2 = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a3 (asing a3) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal3 a2) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (asm (apow a2) (asing a1)) → aal4 (apow a2) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aun (asing a1) (asing (asing a1))) → aex3 (asm (apow a2) (asing a1)) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is 958767... and proposition id is 131d1b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQNKA7G5mfEqmAzZEUVSsxcagnaPsmvBuF)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a1 a2 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a3 a2) → ain a1 (asing a1) → (¬ asubq a1 (asing a1)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ asubq a2 (asing a1)) → (¬ (a1 = apow (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ (a2 = asm (apow a2) a2)) → asubq a3 (apow a2) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (apow (apow (asing a1))) → ain a1 (apow (apow (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → ain a2 (asm a4 (asing a1)) → ain a0 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain a0 (asing a3)) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun a1 (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 (asing (asing a1))) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm a4 (asing a3)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ asubq a3 (asing a2))
In Proofgold the corresponding term root is 2b0ff5... and proposition id is fe59f2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHwigqTtqrB46A8o7iqe4BSBeDbuzY7c8Z)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → ain a0 a2 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ (a1 = a3)) → ain a0 (apow a2) → asubq (asing a1) a2 → (¬ (a0 = asing a2)) → asubq a1 (asm a3 (asing a1)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (apow a2) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ asubq (apow a2) (asing a2)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain (asing a1) (asm a4 a2)) → ain a2 (asm a4 (apow (asing a2))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → (¬ ain (asing a2) (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (asm (apow a2) (asing a2)) → aal3 (aun a2 (asing (apow a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 a3 → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ asubq a2 a1)
In Proofgold the corresponding term root is 3442b3... and proposition id is faef7c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZ4jiW4jyWfFwfAwfQbM5L8pvFr25dKLYt)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (asing a1) → ain a0 (asm a3 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → ain a2 (asm (apow a2) a2) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (asing a1 = asing a2)) → (¬ (asing a1 = a3)) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (apow (asing a1) = a3)) → (¬ (asm a3 (asing a1) = a3)) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a2 (asing a3)) → ain a3 (asing a3) → ain a3 (asm a4 (asing a2)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a2 (asm a4 (apow (asing a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a0 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 a2) (asing a3) = asing a2) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aun a1 (asing a3)) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing (asing a1))) a4) → asubq (apow a2) (aun (apow a2) (asing (apow a2)))
In Proofgold the corresponding term root is b26493... and proposition id is 887c4c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLL5h253Sn8rsoEo8RD6Cz6HNhgfabFYH8)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a0 (asing (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ (a3 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a0 (asm a4 (asing a1)) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a1 (apow (asm a3 (asing a1))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain (apow a2) a4) → ain a2 (asm a4 a2) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (asing (apow a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 a3) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → aal2 (asm a4 a2) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (asm a4 (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → ain a3 (aun (asing a1) (asing a3))
In Proofgold the corresponding term root is 9543e0... and proposition id is 6f370b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFuod3uvMfpd7QNjsdCynETFUQdDuQZ55S)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → (¬ ain a2 a2) → (¬ asubq a4 a3) → (¬ (a1 = a2)) → (¬ ain (asing a1) a0) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (asing a1 = apow a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (apow (asing a1)) a3) → (¬ (a1 = asm (apow a2) a2)) → asubq a3 (apow a2) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun (asing a2) (apow (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (apow a2) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a3 (asm a4 (apow (asing a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a0 (aun a1 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm a4 (asing a3)) a3 → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 a1) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal5 (aun a4 (asing (apow a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asm a3 (asing a1)) a4
In Proofgold the corresponding term root is babfb6... and proposition id is e9187a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSu1M9BF2juXA1z84fJV3hoXWPok59HiTZ)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a1 a1) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain a1 (asm (apow a2) (asing a2)) → (¬ (a0 = asing a2)) → asubq a1 (apow (asing a1)) → ain a2 (asm a3 (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ ain (asing a1) (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a0 (aun a1 (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal2 (asm a3 (asing a1)) → aal3 (aun a2 (asing (apow a2))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1))))
In Proofgold the corresponding term root is ee598e... and proposition id is 26a1c2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVKuLrS3TVvof7zpQMvADWAS6kDWtjYBnw)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (¬ (a1 = a2)) → ain (asing a1) (apow a2) → (¬ ain a0 (asing (asing a1))) → ain a2 (asm (apow a2) (apow (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ asubq a3 (asing a2)) → (¬ asubq (apow a2) (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (apow a2) (apow (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (apow (asing a2))) → (¬ aal4 a3) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 (apow (asing (asing a1))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (apow a2) (asing (apow a2))) → aal5 (aun a4 (asing (apow a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (apow a2) (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1))))
In Proofgold the corresponding term root is 05b5a5... and proposition id is 405af3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMY8g2KJecLppcwqysaGKapxKViczt6rJ29)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → asubq a2 a3 → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain (asing a1) (asm (apow a2) a2) → (¬ ain (asing a1) a1) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → ain a2 (asm (apow a2) (asing a1)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a2) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ ain a3 (apow a2)) → (¬ ain (apow a2) (asing a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a3 (asing a3) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a0 (asm a4 a2)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq a3 (asm a4 (asing a3)) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal2 a2 → aal3 (aun (asing a2) (apow (asing a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 7de748... and proposition id is 549930...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVYrTYuATAd3JYaWF3arWj82786jHdntX9)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a0 a2 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ (a0 = a3)) → ain a2 (asing a2) → (¬ asubq a1 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (apow a2) → (¬ ain a1 (asing a2)) → (¬ (a0 = asm a3 (asing a1))) → ain a2 (apow a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asing a1) a4) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (apow a2) (asing (asing a1))) a3 → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → aal2 (apow (asing (asing a1))) → aex2 a2 → aex2 (asm a3 (asing a1)) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (asm a4 (asing a1)) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))))
In Proofgold the corresponding term root is 086ea5... and proposition id is cbe591...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTa2P1TEGQE3PpivdX8E3pVAPfyMmUv2an)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → ain (asing a1) (asing (asing a1)) → (¬ ain a0 (asing a1)) → (¬ asubq a1 (asing a1)) → (¬ (a0 = asing a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq a2 (asing a2)) → (¬ ain a3 (asing a1)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asing a2) a3) → (¬ ain (asm (apow a2) a2) a3) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain a0 (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain a2 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → ain a3 (asm a4 (asing a2)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3)
In Proofgold the corresponding term root is 9e4c03... and proposition id is e78e4b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHXJuWyP1FshnY7eg364iWPCm2hLRPdpab)
(∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a2 a1) → ain a1 (asing a1) → (¬ ain (asing a1) a2) → (¬ ain a1 (asing a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq (asing a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing a2)) → (¬ (a2 = asing a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a0 (asing a3)) → ain a2 (asm a4 a2) → (¬ ain a3 (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 a2) (asing a2) = asing a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal3 (asm a4 (asing a2)) → aal4 (apow a2) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun (asing (asing a1)) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun a1 (asing a3)) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing (asing a1))) → aex5 (aun (asing (asing (asing a1))) a4) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2)
In Proofgold the corresponding term root is c4e6f1... and proposition id is 9d5450...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJxq59ytQirsz1uUVsPxtREJZAbosa2E1b)
ain a1 a4 → ain a3 a4 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a1) → (¬ (a0 = a2)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain (asing (asing a1)) a2) → (¬ asubq (asing a2) a2) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain a0 (asm a4 a2)) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (asm (apow a2) (asing (asing a1)) = a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal3 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 a4) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 a2 → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex5 (aun (asing (asing a2)) a4) → aex5 (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3)
In Proofgold the corresponding term root is 2e5442... and proposition id is d7eda1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWpm6TG7aHVNDFfuaQxYDL317H4nVN6z4p)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (¬ (a0 = a2)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a0 (asm a3 (asing a1)) → (¬ (a0 = asing a2)) → ain a2 (asm a3 (asing a1)) → ain a2 (asm (apow a2) a2) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ (asing a2 = a3)) → asubq a3 (apow a2) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain a2 (asing a3)) → ain a0 (aun a1 (asing a3)) → (¬ ain a2 (aun a1 (asing a3))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 a1) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aun a1 (asing (apow a2))) → aex3 (asm (apow a2) (asing a1)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is fb1440... and proposition id is 959462...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVJhZYQPyaNHJtwuYtefJwFFG5MMC4fdDS)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ (asing a1 = a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ (a2 = asing a2)) → (¬ (a1 = asm (apow a2) a2)) → asubq a3 (apow a2) → (¬ (asm (apow a2) a2 = apow a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a3 (asm a4 a2) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a1 (aun a3 (asing (apow a2))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm a2 (asing a1)) a1 → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (asm a3 (asing a1)) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (¬ aal4 a3) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (asm (apow a2) a2) → aal2 (apow (asing (asing a1))) → aal3 (asm a4 (asing a1)) → aex2 (apow (asing a1)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex5 (aun (apow a2) (asing (apow a2))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3))
In Proofgold the corresponding term root is b16312... and proposition id is 95c8bc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFApVa1dX8r1qpHs7rU1kSZCw8uGKsuF8a)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (¬ asubq a3 a2) → asubq a3 a4 → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (¬ ain (asing a1) a0) → (¬ (a0 = asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (apow a2) (apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → asubq a3 (apow a2) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asing a1) a4) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a2 (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal2 (asm (apow a2) a2) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 a2 → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25))))
In Proofgold the corresponding term root is efe904... and proposition id is 6adf91...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMT9ySvCZSk6Lw5vtTHmhPrBrKXz1w975GY)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → ain a2 a3 → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (¬ ain a1 a0) → (¬ ain a2 a1) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ asubq a1 (asing a2)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asing a2)) → (¬ ain a3 (asing a1)) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (apow a2) a3) → (¬ ain a0 (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a2 (aun a1 (asing a3))) → ain a1 (asm a4 (asing a2)) → ain a3 (asm a4 (asing a2)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a0 (aun a1 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal5 (aun (asing (asing a2)) a4) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 a2 → aex2 (asm a4 a2) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2)))
In Proofgold the corresponding term root is 8df4ca... and proposition id is c26be4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMU6KgmoVMjikAUNykngBdUjsutzQEieDwz)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ ain (asing a1) a0) → (¬ asubq a1 (asing a1)) → (¬ (a1 = asing a1)) → (¬ ain a0 (asing (asing a1))) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain a2 (asing (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → asubq a3 (apow a2) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ (asm a3 (asing a1) = apow a2)) → ain a1 (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a3 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a1 (aun a3 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → asubq (asm a3 (asing a2)) a2 → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun a4 (asing (apow a2))) → aex3 a3 → aex2 (asm a3 (asing a2)) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a0)) → aex5 (aun (apow a2) (asing (apow a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 a3
In Proofgold the corresponding term root is 13f37b... and proposition id is 8b1de7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQz9NsJJVLsmRT69qjFcncWuacMC1rbghr)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a3 a3) → asubq a1 a2 → (¬ (a0 = a2)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain (asing a1) (asing (asing a1)) → ain (asing a1) (asm (apow a2) a2) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ ain (apow a2) (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing a2) a3) → (¬ (asm a3 (asing a1) = a3)) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a1 (aun (asing a1) (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aex2 (asm (apow a2) a2) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (asm (apow a2) (asing (asing a1))) → aex3 (asm a4 (asing a3)) → aex5 (aun (asing (asing a1)) a4) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1))
In Proofgold the corresponding term root is 26fc72... and proposition id is e7c413...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFWaTXjMEPvKzRf2xN8YLYRLk5oFLhkQcQ)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → asubq (asing a0) a1 → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → asubq a2 a3 → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain (asing a1) a0) → ain a2 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (apow a2) a3) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a0 (apow (asing (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (asm a2 (asing a1) = a1) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (asm (apow a2) (apow (asing a1))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq a1 (asing a1))
In Proofgold the corresponding term root is 4c5a91... and proposition id is 10ce05...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXwQEjH7zqbNWKwV4yiC5GuTM2pHBgms76)
(∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → asubq (asing a0) a1 → (a1 = asing a0) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ asubq a3 a2) → (¬ asubq a4 a3) → (¬ asubq a1 a0) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a0 (asm a3 (asing a1)) → (¬ asubq a1 (asing a2)) → ain a0 (apow (asing a2)) → (¬ (a1 = asing a2)) → (¬ ain a1 (asing (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (apow a2) a2) → (¬ asubq (asm (apow a2) a2) a2) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (apow a2) (asing (asing a1))) a3 → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (apow (asing (asing a1))) → aal3 (asm a4 (asing a1)) → aal3 (aun a2 (asing (apow a2))) → aal5 (aun (asing (apow (asing a1))) a4) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 a4 → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ ain a1 (asing a2))
In Proofgold the corresponding term root is b5b932... and proposition id is 8ad1a1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMM9672Qv237bbV4jqnZbgYbu1mhF746anx)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (¬ asubq a2 a0) → (¬ (a0 = asing a1)) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (a2 = apow a2)) → (¬ asubq (apow a2) (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow a2) a4) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (apow a2) (asing (asing a1)) = a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (aun (asing a2) (apow (asing a2))) → aal3 (aun a2 (asing (apow a2))) → aal4 a4 → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm a4 (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (apow a2) → aex3 (asm a4 (asing a3)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (apow (asing a1))) a4) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3)
In Proofgold the corresponding term root is 095a70... and proposition id is 250580...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHcUtb37aV34wUYBrSUmaptL4jyb94Kw2P)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → asubq a2 a3 → (¬ asubq a3 a2) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (a0 = asing (asing a1))) → (¬ (a0 = asm a3 (asing a1))) → ain a2 (asm a3 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ (a1 = asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (asing a2 = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain a0 (apow (asing (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a3 (asm a4 a2) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq a2 (apow (asm a3 (asing a1))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 a2 → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25)))))
In Proofgold the corresponding term root is 24cdd8... and proposition id is f7cb04...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMakJ3n9sGi7KwCkJqpCAc4T9j8V3JnNRAZ)
ain a0 a2 → ain a1 a3 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ asubq a3 a2) → (¬ asubq a1 a0) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain a0 (asing a1)) → ain a1 (apow a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain (asing a1) a1) → ain (asing a1) (apow (asing a1)) → (¬ ain (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq a2 (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → (¬ asubq a2 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) a3) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ ain (asing (asing a1)) a3) → (¬ (asing a2 = a3)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a2 (asing a3)) → (¬ ain a2 (aun a1 (asing a3))) → ain a3 (asm a4 a2) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a4 (asing a3)) a3 → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 a2 → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (asm (apow a2) (asing a1)) → aal4 a4 → aal5 (aun (asing (asing a2)) a4) → aex2 (aun (asing a3) (asing (apow a2))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a1)) a4) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aal2 (apow (asing (asing a1)))
In Proofgold the corresponding term root is eb3da3... and proposition id is eb6723...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGfsy2udFAxjmHzC9RWE97sBS7u5yfNnY2)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (a0 = asing a1)) → (¬ ain a1 (asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) a3) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a3 (asm a4 (asing a2)) → (¬ ain (asing a1) (asm a4 a2)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (aun a2 (asing (apow a2))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2))
In Proofgold the corresponding term root is ffc258... and proposition id is 4ab052...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKwXr4hjfXY9ppgWP3dU8E3vnucshWwcda)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a1 a0) → asubq a3 a4 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a3 (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (apow a2) (asing (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (a1 = asm (apow a2) a2)) → (¬ (a3 = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain a0 (asm a4 (asing a1)) → (¬ ain (asing a1) a4) → (¬ ain (asing a2) a4) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (apow a2) a4) → ain a2 (asm a4 a2) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm a2 (asing a1)) a1 → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a3 (asing a1)) a4 → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm (apow a2) a2) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a2)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (apow a2) (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ ain a0 (asing (apow a2)))
In Proofgold the corresponding term root is 3e53bb... and proposition id is 0ec79d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRngMQFsbC8TYdqaYbjQoNTQCwUDrpfxns)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → asubq a1 a2 → (¬ asubq a1 (asing a2)) → ain a2 (asm (apow a2) a2) → (¬ (a1 = asing a2)) → (¬ ain a3 (asing a1)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (asing a2 = asm a3 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a3 (aun a1 (asing a3)) → ain a0 (asm a4 (asing a2)) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aun (asing a1) (asing (asing a1))) → aex3 (aun a2 (asing (apow a2))) → aex5 (aun (apow a2) (asing (apow a2))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is 40458a... and proposition id is 3d5066...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdDEPmdegT7GhUDGqo4cPsXHPqug6Vsn5x)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a1 a2 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a1 a0) → (¬ ain a3 a3) → (¬ asubq a2 a1) → (¬ ain (asing a2) a1) → (¬ (a0 = asm (apow a2) a2)) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ (a1 = asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (asing a1 = a3)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain a0 (asing a3)) → ain a3 (asing a3) → (¬ ain (asing (asing a1)) a4) → (¬ ain a0 (asm a4 a2)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 a4) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 a3 → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1))))
In Proofgold the corresponding term root is 23cca9... and proposition id is 55e852...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbBowYkycNXrqyhmJTyjNV6e5sMxh6Nmy1)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a3 a3) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ asubq (asing a2) a2) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing (asing a1)) a3) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain a1 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm a3 (asing a2)) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25))
In Proofgold the corresponding term root is d5752d... and proposition id is 397d79...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJtLzu8dL2P8fbC5TZhjsJEjqbNREjE8sC)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → ain a2 a3 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a2 a1) → ain a0 (apow a2) → (¬ ain a1 (apow (asing a2))) → (¬ ain (asing a2) a1) → (¬ ain a3 (asing a1)) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a0 (asm a4 (asing a2)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aal5 (aun (asing (asing a2)) a4) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 a4 → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → ain a1 (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is 63b2b7... and proposition id is 72925e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRroPdwUazWeAZaUHLvdcWPQtu94anArZD)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a0 (asm (apow a2) (asing a2)) → ain a1 (apow a2) → (¬ (a0 = asm a3 (asing a1))) → ain a2 (asm (apow a2) a2) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asing (asing a1))) → (¬ (asing a1 = a3)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain (apow a2) a4) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (asing a1) (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a3 (asing a1)) a4 → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm (apow a2) a2) → aal2 (asm a4 a2) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (asm (apow a2) (asing a1)) → aex3 (asm a4 (asing a2)) → aex3 (asm a4 (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (¬ (asing a2 = asm a3 (asing a1)))
In Proofgold the corresponding term root is 5da80f... and proposition id is 194f0c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJGYhN4vUTWGNHgh3VZNqD622gcdKKG2d8)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a2 a0) → (¬ ain a0 (asing a2)) → ain (asing a1) (apow a2) → (¬ ain a0 (asing (asing a1))) → asubq a1 (apow (asing a1)) → ain a2 (apow a2) → ain a0 (apow (asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asing a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a1 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a0 (aun a1 (asing a3)) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (asm a4 (asing a2)) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm a2 (asing a0) = asing a1) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (asm a4 (asing a2)) → aal3 (asm a4 (asing a1)) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1))))
In Proofgold the corresponding term root is 255c74... and proposition id is 03420a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHbVubSt64JjYSGaAQ6HYJEtypjxKvw1Gi)
(∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a1 = asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ (asing a2 = asm a3 (asing a1))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a2) a4) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a1 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 a1) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow a2) (asing a1)) → aal3 (asm (apow a2) (apow (asing a2))) → aex2 a2 → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (¬ (a1 = apow a2))
In Proofgold the corresponding term root is 64df11... and proposition id is 5ca579...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMb4Bo4HEhqK19JBYFSZQ1VSF3wK5sdMdKB)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a0 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ (asing a1 = asing a2)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (a1 = apow a2)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a0 (asm a4 (asing a2)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a0 (asm a4 a2)) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a0 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 (apow a2) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → (¬ ain (asing a2) a1)
In Proofgold the corresponding term root is 75bf61... and proposition id is e81fb9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYjbTHVr5YhCPshySPvEfJwTZ6rQoujxho)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ asubq a2 a1) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ ain a1 (apow (asing a1))) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ asubq a2 (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ (asing a1 = a3)) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (a1 = apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a3 (asm a4 (asing a1)) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain a0 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm (apow a2) a2) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm (apow a2) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (apow a2) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ ain a3 (asing (asing a2)))
In Proofgold the corresponding term root is 3aa8cb... and proposition id is b1f9a2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TML7jVPxrKWQFw2PqXXqwiqpxSXjVt8wV7o)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → ain a0 a2 → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, aex2 (apow (asing X24))) → ain a1 (asing a1) → (¬ (a0 = asing a1)) → (¬ (a0 = apow (asing a1))) → (¬ ain a2 (apow (asing a2))) → (¬ asubq (asing a1) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → ain a1 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a1 (asing a3)) → ain a3 (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → ain a2 (aun (asing (asing a2)) a4) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (aun (asing a1) (asing (asing a1))) → aex3 (asm (apow a2) (asing a2)) → aex3 a3 → aex2 (asm a3 (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex4 (apow a2) → aex3 (asm (apow a2) (asing (asing a1))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asing (asing a2)) a4) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ ain a1 (apow (asing a2)))
In Proofgold the corresponding term root is 764a7b... and proposition id is 2d724a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJVMZ2REiyBFFoxCuiztqG4bnV6dtc5wNm)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a2 a0) → (¬ (a0 = a1)) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → ain a2 (asm a3 (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asing a2) a3) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) a4) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain a2 (asing a3)) → (¬ ain (asing a2) (asing a3)) → ain a0 (aun a1 (asing a3)) → (¬ ain (asing a2) (aun a1 (asing a3))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → ain a1 (asm a4 (apow (asing a2))) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → (¬ ain a3 (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (asm a2 (asing a0) = asing a1) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 a3 → aal3 (asm a4 (asing a1)) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → ain a1 (asm (apow a2) (asing a2))
In Proofgold the corresponding term root is 0b65d3... and proposition id is dbe405...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWko9PMsiJqXpk8uwz9VebeyZ5zQ4YhX8f)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (¬ ain a3 a3) → (¬ (a1 = a2)) → (¬ (a2 = a3)) → (¬ ain a1 (apow (asing a2))) → (¬ (a0 = asm a3 (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (asing a1 = asm (apow a2) a2)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain a3 (apow a2)) → (¬ ain (asing (asing a1)) a3) → (¬ (asing a2 = asm a3 (asing a1))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ (a3 = apow a2)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a3 (aun (asing a1) (asing a3)) → ain a0 (asm a4 (asing a2)) → (¬ ain (asing (asing a1)) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain a0 (asm a4 a2)) → ain a3 (asm a4 (apow (asing a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) a4 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aex2 a2 → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3))
In Proofgold the corresponding term root is 8d4211... and proposition id is 53459c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMR1NRfz2rXWfUerLqowXD56y5z4XhYeEwc)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → ain a1 a3 → ain a2 a3 → ain a1 a4 → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a3 a3) → (¬ (a0 = a2)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain (asing a1) (asing (asing a1)) → (¬ ain a0 (asing a2)) → (¬ (a0 = asing a2)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asing a2) (apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a2 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a2 (asing a1)) a1 → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 a2) → (¬ aal3 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun a1 (asing a3)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2))
In Proofgold the corresponding term root is d56836... and proposition id is 172bc6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQj6dYX46Xcs78sfnZHBBXUa1sWkuSabjq)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a0 a2 → ain a1 a2 → ain a2 a3 → ain a2 a4 → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (¬ ain a0 (asing a2)) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain a2 (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ asubq a2 (asing a2)) → (¬ (a1 = asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asing a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun a1 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 a2) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (asm a4 a2) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (apow (asing a1))) a4) → aal5 (aun (asing (asing a2)) a4) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 a4 → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (asm a2 (asing a1) = a1)
In Proofgold the corresponding term root is c1a530... and proposition id is a27838...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXceDPPcWm5gPfmuLP8qoPqAXrroCDh4WZ)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → ain a1 (asing a1) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (asm (apow a2) a2)) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (apow a2) (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (apow (asing a1) = a3)) → (¬ (asm a3 (asing a1) = a3)) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain a2 (asm a4 (asing a1)) → (¬ ain (asing a2) (asing a3)) → ain a3 (aun a1 (asing a3)) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (apow a2) (asing (asing a1))) a3 → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → aal2 a2 → aal2 (asm a3 (asing a1)) → aal3 (asm a4 (asing a2)) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (asm a4 (asing a3)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))))
In Proofgold the corresponding term root is 442f0e... and proposition id is 121d38...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMR3ZsGpNkisT1xznPq9LnTosLiJVE2oLRF)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a1 a2 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → asubq a2 a3 → (¬ asubq a3 a2) → (¬ (a1 = a2)) → ain (asing a1) (asing (asing a1)) → ain a0 (apow (asing a1)) → ain a2 (asing a2) → (¬ ain (asing a1) a2) → (¬ (a0 = asing (asing a1))) → ain a2 (asm (apow a2) (apow (asing a2))) → ain a0 (apow (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asing a2) (apow a2)) → (¬ ain (apow a2) (apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asing (asing a1) = a3)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) a4) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain a0 (asm a4 a2)) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a2 (asing a1)) a1 → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (asm (apow a2) a2) → aal3 (asm (apow a2) (asing a2)) → aal3 (asm a4 (asing a2)) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (apow a2) (asing (apow a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing a1) (asing (asing a1))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → ain a2 a3
In Proofgold the corresponding term root is 58493e... and proposition id is b85776...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNsu9Q5YEWUeABWU3GCXw2mxoAGX7rPNrt)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a0 a1 → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (¬ ain a2 a1) → (¬ ain a0 (asing a2)) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain a1 (asing (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ (a2 = asing a2)) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (asm a4 (apow (asing a2))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a1 (aun a2 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (asm a4 (asing a1)) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 a2 → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (apow (asing a2)) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (asm a4 a2) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ aal5 (aun a3 (asing (apow a2))))
In Proofgold the corresponding term root is 7e07f7... and proposition id is 05343d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMaKcT4PXBszjViTZG7gJNUooWmNFkF4ejh)
(∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a1 a1) → (¬ (a0 = a1)) → (¬ (a0 = a2)) → (¬ ain a0 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asing (asing a1)) a2) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) a3) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ (a2 = apow a2)) → (¬ (a1 = apow a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain (asing a2) (apow (asing a2)) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal3 a3 → aal4 (aun a3 (asing (apow a2))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex5 (aun a4 (asing (apow a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → ain a1 (asm (apow a2) (apow (asing a1)))
In Proofgold the corresponding term root is c45e19... and proposition id is ee5333...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMR3u22ChDZcFoE96qth7ornheiqJYr4cCT)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (¬ ain a3 a2) → (¬ ain a3 a3) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (a1 = asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (asing a2) a3) → (¬ (a0 = apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asing a2) a4) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a3 (aun a1 (asing a3)) → ain a1 (asm a4 (apow (asing a2))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain a0 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 a3 → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is 3fd4a4... and proposition id is ddf8b7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWoQaPGkYv4rNQQvoNMVmXcZgDMQhasuXa)
(∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a2 a0) → (¬ (a0 = a1)) → (¬ (a1 = a2)) → (¬ (a0 = a3)) → (¬ (a2 = a3)) → (¬ ain (asing a1) a0) → (¬ (a0 = apow (asing a1))) → ain a2 (asm (apow a2) a2) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asing (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ asubq a3 (asing a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain (apow (asing a1)) a4) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun a1 (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) a4 → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm a4 a2) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ ain a3 (asing a1))
In Proofgold the corresponding term root is 8ad150... and proposition id is 83a010...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXRFnbmKtm1jVVh3kv9BG3HTZHpgyBk6po)
(∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (¬ ain a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ (a0 = apow (asing a1))) → (¬ ain a2 (apow (asing a2))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a2 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a0 = apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a3 (aun (asing a1) (asing a3)) → ain a3 (asm a4 (asing a2)) → (¬ ain (apow a2) a4) → ain a2 (asm a4 a2) → ain a3 (asm a4 (asing a1)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a2)) a4) → ain a1 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (asm a3 (asing a2)) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing a1)) a4) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is c3f385... and proposition id is 8be225...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcPE2NYzFz9cbBxKY1hQxhg9EvdZfvN3TK)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (¬ ain a0 a0) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a0 (asm a3 (asing a1)) → asubq a1 (apow (asing a1)) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (apow (asing a1))) → ain a2 (asm (apow a2) a2) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (asing a1 = asm a3 (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a0 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq a3 (asm a4 (asing a3)) → (asm a4 (asing a3) = a3) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (apow (asing a2))) → (¬ aal4 a3) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing a2)) a4) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25)))))
In Proofgold the corresponding term root is cc3a7b... and proposition id is 3ca4e0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJE5G3LFd2uh6RUAH4qYXrnrrk88t6yUJs)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (¬ asubq a2 (asing a2)) → (¬ ain a3 (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ (a1 = asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asm a3 (asing a1)) a2) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asing (asing a1) = a3)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a1 (apow (apow (asing a1))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (asm (apow a2) a2) → aal2 (apow (asing (asing a1))) → aal3 (asm (apow a2) (asing a1)) → aal4 (aun a3 (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (apow a2) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3))
In Proofgold the corresponding term root is 237c36... and proposition id is 64edfd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWLEewGn9RBdQpAs2fwjQeKdrSbCGneiuR)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (¬ ain a1 a1) → (¬ ain a2 a2) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ ain (asing a1) a0) → ain a0 (apow a2) → asubq (asing a1) a2 → (¬ asubq a1 (asing a2)) → (¬ ain a1 (apow (asing a2))) → (¬ (a0 = asing (asing a1))) → (¬ ain a2 (asing a1)) → (¬ ain a2 (apow (asing a1))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asing a2) a3) → (¬ ain (apow a2) a3) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (asm a3 (asing a1) = a3)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a3 (aun a1 (asing a3)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (apow a2) (asing (asing a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (apow (asing a1)) → aex2 (asm a3 (asing a1)) → aex2 (apow (asing a2)) → aex3 a3 → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (apow a2) (asing a0)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4)
In Proofgold the corresponding term root is 93fd1f... and proposition id is 6c33f1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTbiuzZREH7wA47gSqeeNBsU9boQ3ecuiZ)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a2 a2) → (¬ asubq a2 a1) → (¬ asubq a4 a3) → ain a1 (asing a1) → ain a1 (apow a2) → (¬ ain (asing a2) a1) → (¬ ain a2 (apow (asing a1))) → (¬ ain a3 (asing a1)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a1 = asm (apow a2) a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a1 (aun (asing a1) (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun a2 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → asubq a1 (asm a2 (asing a1)) → (asm a2 (asing a1) = a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow a2) (apow (asing a2))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ aal4 (asm (apow a2) (asing a1)))
In Proofgold the corresponding term root is 073489... and proposition id is 207255...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZQSeg9KUHopUFataB9WL9nPpinThhd6qq)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ asubq a4 a3) → (¬ (a1 = a2)) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain (asing a2) a2) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a2 = apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asing a2 = apow a2)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain a0 (asm a4 a2)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a4 (aun (asing (asing a2)) a4) → (asm a3 (asing a2) = a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal3 (aun (asing a1) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm a4 (asing a3)) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ (a0 = a2))
In Proofgold the corresponding term root is bcb826... and proposition id is 7f2201...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKs6igREbgfU4HeWfVKKDDcpTMametAfKH)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ (asing a1 = a2)) → (¬ ain a2 (apow (asing a2))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a1) (asing a1)) → (¬ ain a3 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (apow (asing a1)) a3) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ ain a0 (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain a3 (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a2 (asm a4 (asing a1)) → ain a3 (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a2 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a1 (aun a3 (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a1) = a1) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a4 (asing a3)) a3 → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (asm a4 (asing a1)) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 a2 → aex3 a3 → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq (apow a2) (aun (apow a2) (asing (apow a2)))
In Proofgold the corresponding term root is f69d18... and proposition id is d93c09...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcUC4G8Cx5qboH7RLiapXNjH9Q3ttj8bfs)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a3 a2) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ (a0 = asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ ain (asing a1) a1) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain a3 (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asm (apow a2) a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ ain a0 (asing (apow (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (apow (asing a2)) → aex3 (asm a4 (asing a3)) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is c65915... and proposition id is 5ed4ea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKRDNmF787B3KGKvTri5Hj3quCDqKJcQL2)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → ain a0 (asm a3 (asing a1)) → asubq (asing a1) a2 → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a1 (asing (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ (asing a1 = apow a2)) → (¬ (a2 = asm (apow a2) a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ (asm a3 (asing a1) = a3)) → (¬ (asing a2 = apow a2)) → ain a0 (apow (apow (asing a1))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a2 (asm a4 (asing a1)) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (asm a3 (asing a2) = a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal5 a4) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (asm a4 (asing a2)) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aun a1 (asing a3)) → aex3 (asm (apow a2) (asing a2)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ asubq a3 a2)
In Proofgold the corresponding term root is 98022e... and proposition id is d128c0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMa6WAEvHEz3eNsq4BhpdaskCdu9JoHCx4Q)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → ain a1 a3 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → asubq (asing a1) a2 → (¬ (a0 = asing a1)) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq (asing a1) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a2 = asm (apow a2) a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → asubq a3 (apow a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a3 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (asm a2 (asing a1) = a1) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal4 a3) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 a4) → (¬ aal6 (aun (asing (asing a2)) a4)) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun (asing (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is 9acf09... and proposition id is 4686be...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXwcg4Zg7sc2hvQfcbjoPKRtAVhyj6vVbp)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → ain a0 (apow a2) → (¬ ain a0 (asing a1)) → (¬ asubq a2 (asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (apow a2) (apow a2)) → (¬ ain (asing (asing a1)) a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a1 (asing a3)) → ain a1 (aun (asing (asing a2)) a4) → ain (asing a2) (aun (asing (asing a2)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (asing (apow a2)) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a4 (asing a3)) a3 → asubq a3 (asm a4 (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal3 a2) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal2 a2 → aal2 (apow (asing (asing a1))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun a3 (asing (apow a2))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (apow (asing a2)) → aex2 (asm a3 (asing a2)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2)
In Proofgold the corresponding term root is d3b50e... and proposition id is 17bc63...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQyeBdyXDy74W5WcLHEhvSrTFHKohCkhcA)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ asubq a3 a2) → ain a0 (apow (asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain (asing a1) (apow a2) → (¬ ain (asing a2) a1) → asubq a1 (apow (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ (asing a1 = a3)) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asing a2) (apow a2)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (asm a4 a2) → ain a1 (aun a2 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → (asm a4 (asing a3) = a3) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (asm a4 a2) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (asing a2))) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq a4 (aun (asing (asing a1)) a4)
In Proofgold the corresponding term root is 19104c... and proposition id is 8c8ece...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJcFvFEaf4cvZBrdeH47Qwf7aFeV5VVrw9)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ (a1 = a2)) → (¬ asubq a2 a0) → ain a0 (apow (asing a1)) → ain (asing a1) (apow a2) → ain a1 (asm (apow a2) (asing a2)) → (¬ (asing a1 = a2)) → (¬ asubq a2 (asing a2)) → ain a0 (apow (asing a2)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (apow a2) (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain a0 (apow (apow (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (asing a1)) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm a3 (asing a2)) a2 → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 a3 → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 a2 → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm (apow a2) a2) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (aun a1 (asing (apow a2))) → aex2 (aun (asing a3) (asing (apow a2))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (asing a1) (asm (apow a2) (asing a2))
In Proofgold the corresponding term root is 4f0402... and proposition id is e54a48...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbaV5dxFkc97kJtdMRLga4MM7KkBr377pJ)
ain a0 a4 → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a2 a1) → ain (asing a1) (apow (asing a1)) → ain a2 (asm (apow a2) a2) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → aal2 (asm a4 a2) → aal3 (asm a4 (asing a1)) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is f2e81d... and proposition id is bc7113...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbfH9pgtP49ZT8VRGt3Z8dNvZkxKHRhr9Q)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ (a1 = asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (apow a2) a2) → (¬ asubq (asing a2) a2) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain a1 (asm a4 (asing a2)) → (¬ ain (apow a2) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm (apow a2) a2) → aal2 (apow (asing (asing a1))) → aal3 a3 → aal4 (aun a3 (asing (asing a2))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (apow (asing a2)) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (asm a4 (asing a2)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex3 (asm a4 (asing a3)) → aex5 (aun (asing (asing a1)) a4) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1)))
In Proofgold the corresponding term root is 6f3ec6... and proposition id is 9637b9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFG2E4cayoQdpVtEYL1jgX6DLWeSb61tUL)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a0 a3 → ain a0 a4 → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ ain a3 a3) → (¬ ain a0 (asing (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a3) → (¬ (a1 = asm (apow a2) a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → asubq a3 (apow a2) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a3 (asm a4 a2) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (asing (apow a2)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (asm a2 (asing a0) = asing a1) → (asm a2 (asing a1) = a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (apow a2) (asing (asing a1))) a3 → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) a4 → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (asm a4 a2)) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (apow (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (apow a2) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ ain (asing a2) a3)
In Proofgold the corresponding term root is 86af97... and proposition id is 4d24e1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKe6XZxevien8pLbWnvF9ULnFp8LBsS29q)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (¬ (a0 = asing (asing a1))) → (¬ ain a1 (asing a2)) → (¬ (a1 = apow (asing a1))) → (¬ ain (asing a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asm (apow a2) a2) a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a2 (asm a4 (apow (asing a2))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 a4 → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (asm (apow a2) (asing (asing a1))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → asubq a1 (apow (asing a1))
In Proofgold the corresponding term root is 7e3e1d... and proposition id is 96c2fd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUTLKBKmiqyNF8VWvZ2z5tthYqAc5kxDVu)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a3 a2) → (¬ ain a0 (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ (a1 = apow (asing a1))) → (¬ (a2 = asing (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asm a3 (asing a1))) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain a1 (apow (asm a3 (asing a1))) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun a1 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a4 (aun a4 (asing (apow a2))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal4 (apow a2) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (asm a3 (asing a1)) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → ain a1 (aun (asing a1) (asing a3))
In Proofgold the corresponding term root is 120645... and proposition id is 0834b8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZeKYRgCtkShpu7thReybEc82F4Y1Rnphr)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → ain a1 a2 → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (¬ ain a2 a0) → (¬ ain a2 a1) → (¬ asubq a3 a2) → (¬ (a0 = a2)) → (¬ (a2 = a3)) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ (a1 = apow (asing a1))) → (¬ ain (asm (apow a2) a2) a3) → asubq a3 (apow a2) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a3 (asm a4 (asing a1)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) a4) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq a2 (asm a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal4 (apow a2) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 a3 → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24))))
In Proofgold the corresponding term root is f78b62... and proposition id is 5da3b6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZ3CBa1rZNn1tduqEnGg4j1SwKEDqA9NvG)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → asubq a2 a3 → ain a2 (asing a2) → ain a1 (asm (apow a2) (apow (asing a1))) → asubq a1 (asm a3 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (asing a1) (asing a1)) → (¬ (a1 = asing a2)) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a1 (asing a3)) → ain a2 (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain a0 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asing a1) (asm a2 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (asm (apow a2) (apow (asing a1))) → aex3 (asm a4 (asing a2)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2)))
In Proofgold the corresponding term root is 96dcf9... and proposition id is 9b9875...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMn26K4jGMctdz2duaj7a9t4y8SD4ze331)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a2 a0) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain a1 (asing a2)) → (¬ ain a2 (apow (asing a1))) → ain a0 (apow (asing a2)) → (¬ asubq (asm a3 (asing a1)) a2) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (asing a2 = a3)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a0 (asing a3)) → (¬ ain a2 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal3 a2) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (apow (asing a2))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm (apow a2) a2) → aal3 a3 → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (apow (asing a2)) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex4 (aun a3 (asing (apow a2))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24)
In Proofgold the corresponding term root is c5462e... and proposition id is ae1349...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTFUdWYE8ZJCbGGEL7a5mmjtpu4tu93swQ)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a1 a2 → ain a3 a4 → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a1 a1) → (¬ ain a2 a1) → (¬ (a0 = a3)) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (¬ ain a0 (asing a2)) → (¬ ain a1 (apow (asing a1))) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a0 (apow (asing a2)) → (¬ ain a1 (asing (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ ain (apow a2) a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → ain a0 (apow (asing (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a0 (asm a4 (asing a1)) → ain a2 (asm a4 (asing a1)) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a1 (aun (asing a1) (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a1 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (asm a4 (asing a2)) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun a4 (asing (apow a2))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm a4 (asing a3) = a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 a1) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm a3 (asing a1)) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))))
In Proofgold the corresponding term root is 8fe52c... and proposition id is 6d5ce2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTF7vwrgfYc6SYeWg2dSHUZppRZmyXrmn6)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (asing a1) → ain a1 (aun (asing a1) (asing (asing a1))) → ain (asing a1) (apow a2) → (¬ ain a1 (asing a2)) → asubq a1 (apow (asing a1)) → (¬ ain a2 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain a3 (asing a2)) → (¬ (a2 = apow a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ (asing (asing a1) = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a3 (asing a3) → (¬ ain a2 (aun a1 (asing a3))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (asm a4 (asing a3) = a3) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal2 (asm a4 a2) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (asm a4 (asing a1)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 a2)
In Proofgold the corresponding term root is 1d0691... and proposition id is 5e5e3c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLshzS2eG7Z6e6SzF8Le3hQGPSYiNxJGVX)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a1 a2 → ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a3 a2) → (¬ (a1 = a2)) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ (a0 = apow (asing a1))) → ain a2 (apow a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asm (apow a2) a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asm (apow a2) a2 = apow a2)) → ain a1 (apow (apow (asing a1))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a3 (asing a3) → ain a3 (asm a4 (asing a2)) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (asing a1) (asm a4 a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a0 (aun a1 (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (apow a2) (asing (asing a1))) a3 → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (¬ aal4 a3) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun a3 (asing (asing a2))) → aex3 (asm a4 (asing a3)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2))))
In Proofgold the corresponding term root is b62f67... and proposition id is 795837...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYiuQmka6sJ7SKvatzny9DtMD8HCc9uGUC)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → asubq a1 a2 → asubq a2 a3 → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a0 (asm (apow a2) (asing a2)) → ain a0 (asm a3 (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ (a2 = asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing a2) a3) → (¬ (a1 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a2) (asing (asing a2)) → ain a3 (asing a3) → ain a0 (aun a1 (asing a3)) → ain a3 (aun a1 (asing a3)) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 a3) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (apow (asing a2)) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (asm a4 (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)))
In Proofgold the corresponding term root is 3161c5... and proposition id is 0b3c09...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdqeeguJ28c2Z73KZE9L7Mpg2wDj64ngHK)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a1 a0) → (¬ (a1 = a2)) → (¬ ain a2 (apow (asing a1))) → (¬ (a1 = asing a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (asing a2) a3) → (¬ ain (asm (apow a2) a2) a3) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a3 (aun a1 (asing a3)) → ain a3 (aun (asing a1) (asing a3)) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a2 (asing (apow a2))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 a3) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1)))
In Proofgold the corresponding term root is 7c990d... and proposition id is 9a0d5a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdxGy6JkSAByyfuaqwHcA41Q5vrRyr3HWW)
(∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a1 a0) → (¬ ain a1 a1) → (¬ ain a3 a3) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a0 (apow (asing a1)) → ain a0 (apow a2) → (¬ ain (asing a1) a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ asubq a1 (asing a2)) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (asing a2)) → ain a0 (apow (apow (asing a1))) → ain a1 (apow (apow (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain (asing a2) (asing a3)) → ain a0 (asm a4 (asing a2)) → ain a3 (asm a4 (asing a2)) → (¬ ain a3 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun a3 (asing (asing a2))) → aex2 (asm a4 a2) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25)))
In Proofgold the corresponding term root is 5a280e... and proposition id is b52285...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHMj7zr9dEdgqANMHNo9uhLWTFcngoXoFK)
ain a2 a3 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ ain a0 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain (asing a2) a1) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq a2 (asing a1)) → (¬ ain a3 (asing a1)) → (¬ asubq (asing a1) (asing a2)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ (a2 = apow (asing a1))) → (¬ ain (asing (asing a1)) a3) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (apow (asing a2)) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a3 (asing a3) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (apow a2) a4) → ain a3 (asm a4 (apow (asing a2))) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a1) (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal2 (asing (asing a1))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal2 a2 → aal3 a3 → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (asm a3 (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (¬ ain (asing a1) (asm a4 a2))
In Proofgold the corresponding term root is d3dd16... and proposition id is e31628...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJRWJ8V3Jm949Jt2wbeEbcMyn7TkqLuhmB)
(∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a0 a3 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a0 a0) → ain a2 (asm (apow a2) a2) → ain a0 (apow (asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ (a1 = apow (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (asm a3 (asing a1) = a3)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (a3 = apow a2)) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain (asing a2) (apow (asing a2)) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (asing a2) a4) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a0 (asm a4 (asing a2)) → (¬ ain (asing (asing a1)) a4) → ain a1 (asm a4 (apow (asing a2))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq a3 (asm a4 (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm a3 (asing a2)) → aex3 (aun (asing a2) (apow (asing a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1)))
In Proofgold the corresponding term root is 77515f... and proposition id is 5e4c12...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPLKn5W9rFtX7xPNfnxwsHuhpP2wtxziwv)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → asubq a1 (asing a0) → (a1 = asing a0) → (¬ ain a3 a2) → asubq a3 a4 → (¬ ain a0 (asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ (a0 = asing (asing a1))) → (¬ asubq a2 (asing a2)) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (asing (asing a1) = a3)) → (¬ (asm a3 (asing a1) = a3)) → (¬ ain (asing a2) (asm (apow a2) a2)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (asm a4 (asing a1)) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain (asing a2) (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow a2) (asing a2)) → aal3 (asm (apow a2) (apow (asing a2))) → aal5 (aun (asing (asing a1)) a4) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (¬ ain (asing a2) (asm a3 (asing a1)))
In Proofgold the corresponding term root is e13d93... and proposition id is 798bd8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUegnTqSQu6ai4jkFG1Th7FjSSs3qxpwDr)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a0 a0) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a1 (apow a2) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asing a2)) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain a2 (apow (asing a2))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a2) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ ain (asing a1) a3) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ (a2 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ ain (asing a2) (asm (apow a2) a2)) → ain a0 (apow (asing (asing a1))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal5 (aun (asing (asing a1)) a4) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm a4 a2) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing (asing a1))) a4) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is ec67d3... and proposition id is 1980fa...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGdcXiaSH9yF6v4MTA3QA6SetbBara42E9)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a2 a0) → (¬ asubq a1 a0) → ain a2 (asing a2) → (¬ ain (asing a1) a2) → ain (asing a1) (asm (apow a2) a2) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (a0 = asing (asing a1))) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (asing (asing a1)) a3) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asm (apow a2) a2) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a1 (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing a2)) a4) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (asing (asing a1))) a3 → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing (asing a1))) → (¬ aal4 (asm a4 (asing a1))) → aal3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))
In Proofgold the corresponding term root is ddeb35... and proposition id is 5ef817...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdspvfVrX3WsbXvYRuXnWJThFKdTz8qt5A)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a2) → (¬ asubq a3 a2) → (¬ asubq a4 a3) → (¬ (a2 = a3)) → ain a2 (asing a2) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ ain (asing a1) a2) → ain a2 (apow a2) → (¬ ain a3 (asing a1)) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (asing a2) a3) → asubq (asm a3 (asing a1)) (apow a2) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a2 (aun a1 (asing a3))) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun a3 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → aal3 (asm a4 (asing a2)) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (aun (asing a2) (asing (asing a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (¬ ain a3 (apow a2))
In Proofgold the corresponding term root is 44610d... and proposition id is e8dfa2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TML31nJVUBU9P5iZ1qaB5vpuUT8uA7gGtYF)
ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ ain a2 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (apow a2) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ ain (apow a2) a3) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a3 (aun a1 (asing a3)) → ain a1 (asm a4 (asing a2)) → adisjoint (apow (asing a1)) (asm a4 a2) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal3 (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 a3 → aal3 (aun (asing a2) (apow (asing a2))) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun a1 (asing (apow a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))
In Proofgold the corresponding term root is 9c7daa... and proposition id is a76eef...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMM61w67AkjJw1y58cKHBLNbntwWXF24rWT)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a1 a2 → ain a0 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a2 a2) → asubq (asing a1) a2 → ain a2 (asm a3 (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a2) (apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = asm a3 (asing a1))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a3 (asing a3) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal3 (asm (apow a2) (asing a2)) → aal4 (apow a2) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asing (asing a2)) a4) → aex2 a2 → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1))
In Proofgold the corresponding term root is a865fa... and proposition id is 779914...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMb7YgX7tJ5cRNwdbNGjzhiqWjTMWFmbgwK)
(∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a1 a2 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (¬ (a0 = a1)) → (¬ (a0 = asing (asing a1))) → (¬ ain (asing a1) (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain a3 (asm a3 (asing a1))) → ain a0 (apow (asing (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain a2 (aun a1 (asing a3))) → ain a3 (asm a4 a2) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm a3 (asing a1)) a4 → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal5 (aun (asing (asing a1)) a4) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (apow (asing a2)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))
In Proofgold the corresponding term root is e4c152... and proposition id is a17bf5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTDd3hNsK1Sm6VmmrpjH4Wmp6Pch6PyxYw)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (¬ ain (asing a1) a0) → (¬ (asing a1 = a2)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ asubq (asing a1) (asing (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asing a1) (asm a2 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (apow (asing a1)) → aex2 (asm a4 a2) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aun a2 (asing (apow a2))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun a4 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))))
In Proofgold the corresponding term root is 72344c... and proposition id is 4ea3fc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJ7NTWpwpEUFXSPrT8thC6qg2BXREkbbFU)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ ain (asing a1) (asing a2)) → (¬ (a1 = asm a3 (asing a1))) → (¬ (asing a1 = asm a3 (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ (a2 = asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = a3)) → asubq a3 (apow a2) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a1 (apow (apow (asing a1))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a0 (aun a1 (asing a3)) → ain a3 (asm a4 (apow (asing a2))) → ain a1 (aun a3 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing a2)) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aun a1 (asing (apow a2))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1)))
In Proofgold the corresponding term root is 871cee... and proposition id is a59f11...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNJYV98YoTsLwntgp9wMRGNHM1EPd4VLTE)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ ain a2 a0) → (¬ ain a2 a1) → (¬ ain a1 (apow (asing a2))) → (¬ (a0 = asing (asing a1))) → (¬ ain a1 (asing a2)) → ain a2 (apow a2) → asubq (asing a1) (asm (apow a2) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (asing a2 = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain a0 (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → (asm a2 (asing a0) = asing a1) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 a2 → aal2 (asm a3 (asing a1)) → aal3 (asm (apow a2) (asing a1)) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (aun a2 (asing (apow a2))) → aex4 (apow a2) → aex3 (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ ain a3 (asm (apow a2) (asing a1)))
In Proofgold the corresponding term root is dea22c... and proposition id is 7d85c9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLquGZFn7yhRZqpe49qaAYyM1PFRSi8DpC)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → ain a0 (asm a3 (asing a1)) → asubq (asing a1) a2 → (¬ ain a1 (apow (asing a2))) → (¬ (asing a1 = a2)) → (¬ asubq a2 (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (apow a2) (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (asing a2 = a3)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain a1 (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → ain a3 (asm a4 (asing a2)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (asm a4 a2) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (asm a4 a2) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aun a3 (asing (asing a2))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26))
In Proofgold the corresponding term root is fdaee9... and proposition id is 047d28...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbUUj8UAtZtvj7xfVkfJyApH1HNBnTtDN4)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ ain a2 a0) → (¬ ain a3 a3) → asubq a2 a3 → (¬ asubq a1 (asing a1)) → (¬ (a0 = asing a2)) → (¬ (a0 = asm a3 (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (apow a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asing a2)) → (¬ ain (asm (apow a2) a2) a3) → (¬ (apow (asing a1) = a3)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a0 (apow (asing (asing a1))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain (apow (asing a1)) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a2 (asm a4 a2) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a4 (asing a3)) a3 → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 a4 → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (apow a2) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2)))
In Proofgold the corresponding term root is 4b3760... and proposition id is 677f51...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWdSPAyy5yutSf2iCSJxpXQXrSWkQut1ps)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ asubq a2 a0) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ (a0 = asm a3 (asing a1))) → (¬ (a1 = asm a3 (asing a1))) → (¬ (asing a1 = a3)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → ain a3 (asing a3) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (asing (apow a2)) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal2 (asing a3)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing a1) (apow (asing a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → aal5 (aun (asing (asing (asing a1))) a4)
In Proofgold the corresponding term root is 0b648c... and proposition id is ea8ff7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMby3LETewPfHyQYcdGWRXPkmR479kA9Kmd)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a1 (asing a1) → (¬ ain a0 (asing a2)) → (¬ ain a0 (asing a1)) → (¬ asubq a1 (asing a2)) → (¬ ain (asing a1) a1) → ain (asing a1) (apow (asing a1)) → (¬ ain a2 (apow (asing a1))) → (¬ ain (asing a1) (asing a1)) → (¬ (a1 = asm a3 (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a0 = apow a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a3 (aun a1 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq a3 (asm a4 (asing a3)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (¬ aal2 (asing a2)) → (¬ aal3 a2) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (asm a4 (asing a2)) → aex2 (asm a3 (asing a1)) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm a3 (asing a2)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)))
In Proofgold the corresponding term root is 0ff3f4... and proposition id is ef574a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQRuotbw7AZtWbFzFvePicJJjrShupXRX5)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a2 a0) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asm (apow a2) a2)) → asubq a1 (apow (asing a1)) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ asubq (asing a2) a2) → (¬ asubq (asm a3 (asing a1)) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (a0 = apow a2)) → (¬ ain (apow a2) a3) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a1 (apow (apow (asing a1))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 a4) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (aun a1 (asing a3)) → aex2 (aun a1 (asing (apow a2))) → aex4 a4 → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 9cd1dc... and proposition id is 837ca0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQ7nFYE9ZMDeKa9NPHnYQLMYK2MuS9C1V3)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (¬ (a0 = a1)) → (¬ (a1 = a3)) → (¬ asubq a2 (asing a1)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asing a2) (aun a1 (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (asm a4 a2) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a1 (aun a3 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (apow (asing a1))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal2 (asm a4 a2) → aal3 a3 → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal5 (aun (apow a2) (asing (apow a2))) → aex3 (asm (apow a2) (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (asm a4 (asing a0)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))))
In Proofgold the corresponding term root is 0a15ff... and proposition id is 8b8e9f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYi8VX6WWMPnbr3TZpGmJvexghBQDTYHp2)
ain a1 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a0 a0) → (¬ (a1 = a3)) → ain (asing a1) (asm (apow a2) a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = apow (asing a1))) → (¬ asubq (asing a1) (asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ ain (apow a2) a3) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asm (apow a2) a2 = apow a2)) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain a0 (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal3 (asm a4 a2)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (apow a2) → aex3 (asm a4 (asing a0)) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))
In Proofgold the corresponding term root is f9cdfc... and proposition id is 0a4c0c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVkthpDTMP69aAabXYWps51LXTEHuNp5tN)
ain a0 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (asing (asing a2)) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal5 (aun a3 (asing (asing a2)))) → aal2 (asm a3 (asing a1)) → aal3 (aun a2 (asing (apow a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun (asing (asing a1)) a4) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (apow a2) (asing (asing a2))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ ain (apow a2) (apow (asing (asing a1))))
In Proofgold the corresponding term root is b63dfa... and proposition id is 476cb3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKjS7ETQyXRmURRsJQMyovvh1Y1BYPbfn7)
ain a1 a2 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (¬ ain a1 a0) → (¬ (a1 = a3)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ asubq a1 (asing a1)) → (¬ (a0 = asm (apow a2) a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain (asing (asing a1)) a2) → (¬ asubq (asing a2) a2) → (¬ ain a2 (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ (a0 = apow a2)) → (¬ (asing (asing a1) = a3)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a1 (asm a4 (apow (asing a2))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm a2 (asing a0)) (asing a1) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun a1 (asing a3)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))
In Proofgold the corresponding term root is a9a288... and proposition id is cd18e4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdSvPT98j1YoJNytuCMBUq2e7jb2fDoyTn)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a2 a3 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a3 a2) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ ain (asing a1) a0) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a0 (apow a2) → (¬ asubq a1 (asing a2)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (a0 = asm a3 (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm (apow a2) a2) a3) → (¬ (asing (asing a1) = a3)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a0 (aun (asing (asing a2)) a4) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asing a1) (asm a2 (asing a0)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (apow a2)) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal6 (aun (asing (asing a2)) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal4 (aun a3 (asing (apow a2))) → aex2 a2 → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1))
In Proofgold the corresponding term root is deb38e... and proposition id is c4aa87...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRGdvdKKQoJHAggMWTpdjspa6C8PcjYBoA)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a0 a4 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a3 a2) → (¬ (a0 = a1)) → (¬ (a1 = a3)) → ain (asing a1) (asm (apow a2) a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a3 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (asing a1) a3) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain (apow a2) (apow a2)) → (¬ (a1 = apow a2)) → (¬ (a3 = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a2) a4) → (¬ ain a2 (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain a1 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm a4 (asing a3) = a3) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (aun a2 (asing (apow a2))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2
In Proofgold the corresponding term root is 8da11e... and proposition id is 8f1c31...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQHEcKGcMMrACbX4VHhCSZXX6gwiq2sTRf)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a1 a2 → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → asubq a2 a3 → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a0 (asm (apow a2) (asing a2)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) a2) → ain a2 (asm a3 (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ ain a2 (apow (asing a2))) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (a2 = apow a2)) → (¬ ain (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (asing a2 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a1 (apow (asm a3 (asing a1))) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (asing (asing a1)) a4) → ain a2 (asm a4 a2) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal2 (asing a2)) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun a1 (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (apow a2) → aex4 (aun a3 (asing (apow (asing a1)))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1))))
In Proofgold the corresponding term root is dcc47a... and proposition id is 987ab7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLrewqJqHYAw7sF4gVvdbKRsgb8op5vEXC)
(∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → ain a0 (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain a0 (asm (apow a2) (asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) a1) → (¬ (asing a1 = a2)) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asing a2)) → (¬ (a2 = asing (asing a1))) → (¬ ain a3 (asing a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ (a3 = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a1 (apow (apow (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain (asing a2) (asing a3)) → ain a0 (asm a4 (asing a2)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → aal2 (asm a3 (asing a1)) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2))))
In Proofgold the corresponding term root is c4aafe... and proposition id is f246d3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMV1G25UnhSF2m6GaEp3LrHM2Et4ww3RyiF)
(∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ asubq a3 a2) → (¬ asubq a1 a0) → (¬ (a0 = asing (asing a1))) → (¬ ain a2 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → ain a0 (apow (asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (a2 = asm (apow a2) a2)) → (¬ (a2 = apow a2)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain a0 (asm a4 (asing a1)) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a1 (apow (asm a3 (asing a1))) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (asing (apow a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 a3) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (aun (asing a2) (apow (asing a2))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (asm a4 a2) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))))
In Proofgold the corresponding term root is e0bfd7... and proposition id is beba76...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKZNTWR91rse1HPeBgFRKzjTZbnVmCK1tr)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ (a0 = a3)) → (¬ (a2 = a3)) → ain a0 (asm (apow a2) (asing a2)) → (¬ asubq a1 (asing a1)) → ain a2 (apow a2) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ asubq (asm (apow a2) a2) a2) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain a1 (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a2) (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → ain a2 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a2)) a2 → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain (asing (asing a1)) (apow (asing (asing a1)))
In Proofgold the corresponding term root is cdc91d... and proposition id is 0d9176...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMW67yu9YHKdHqSejfM4KfwqDyNDZRvoJRt)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a0 a2 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → asubq a2 a3 → (¬ asubq a3 a2) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (a2 = asm a3 (asing a1))) → ain a0 (apow (asing (asing a1))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (asing a1) (asm a4 a2)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a1 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 a4) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (asm a4 (asing a1)) → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25))
In Proofgold the corresponding term root is 52610d... and proposition id is 80c7b9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQgYWrT28AvBDsDBgdt4tEibKiswZw11ds)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → asubq a1 (asing a0) → (∀X24 : set, aex2 (apow (asing X24))) → ain a1 (asing a1) → (¬ ain a1 (apow (asing a1))) → (¬ (a0 = asing (asing a1))) → (¬ (a0 = asm (apow a2) a2)) → asubq a1 (apow (asing a1)) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ (a1 = asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → asubq a3 (apow a2) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a1 (apow (apow (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a2) (asing a3)) → ain a0 (aun a1 (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain (asing a1) (asm a4 a2)) → ain a2 (asm a4 a2) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a1 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (aun (asing a1) (apow (asing a2))) → aex2 (asm (apow a2) a2) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aun a3 (asing (asing a2))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asing (asing a2)) a4) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ (asm (apow a2) a2 = apow a2))
In Proofgold the corresponding term root is 56c4b7... and proposition id is 6a218f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMvokmR5NPu8YYbunss7H6NhV5xN8QDEWS)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a0 a0) → (¬ ain a1 a1) → (¬ ain a0 (asing (asing a1))) → asubq a1 (asm a3 (asing a1)) → ain a2 (apow a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (asing a2 = a3)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a2) a4) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain (apow a2) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → asubq a1 (asm a2 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → aal2 a2 → aal3 (asm (apow a2) (apow (asing a2))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun a1 (asing (apow a2))) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0)))
In Proofgold the corresponding term root is ba89d5... and proposition id is b486d0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTzHiLGZ6cVijXyRqkUuoKSSC4R253HNYk)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (¬ ain a2 a1) → (¬ ain (asing a1) a1) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain (asing a2) a1) → asubq a1 (apow (asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) a3) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a0 (apow (asing (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (asm a4 (asing a1)) → ain a2 (aun a3 (asing (asing a2))) → ain a3 (asm a4 (asing a1)) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a1 (asing (apow a2))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (asm a4 (asing a3) = a3) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (¬ aal2 (asing a1)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (apow (asing a2)) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 a3 → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (apow a2) (asing (apow a2))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ (a3 = asm (apow a2) a2))
In Proofgold the corresponding term root is 6ff552... and proposition id is 9346a5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMXUT5frMrnDqZQVF8SX2CGX9jid4B5y77)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (¬ asubq a2 a1) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ ain a0 (asing a1)) → (¬ ain a0 (asing (asing a1))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = a3)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (asing a2) (aun a1 (asing a3))) → (¬ ain (asing a2) (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (asm a3 (asing a1)) a4 → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (¬ aal3 (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm a4 (asing a1)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (¬ (asm (apow a2) a2 = apow a2))
In Proofgold the corresponding term root is 3ac176... and proposition id is 22a66c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZPL7wUsyxvr3ye8mgy9MMS6yiB9t26Kdu)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → asubq a3 a4 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain (asing a1) (asing (asing a1)) → (¬ ain a1 (apow (asing a1))) → asubq (asing a1) a2 → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (apow (asing a1)) a3) → (¬ (a1 = apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (asing a3)) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (asing a1) (asm a4 a2)) → ain a2 (asm a4 a2) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (asm a4 (asing a3) = a3) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (aun (asing a2) (apow (asing a2))) → aex2 (asm a3 (asing a2)) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (asm a4 (asing a2)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → ain (apow a2) (aun a2 (asing (apow a2)))
In Proofgold the corresponding term root is 615beb... and proposition id is 81c401...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZUuzoALxYvYhscmg413HJgUPUeGqvwgc6)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a0 a0) → (¬ ain a2 a1) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain (asing a1) a0) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ (a1 = asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a1) (asing a2)) → ain a2 (asm (apow a2) (apow (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing a2) a2) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain a3 (apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (asm a3 (asing a1) = a3)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asm a3 (asing a1) = apow a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq a2 (apow (asm a3 (asing a1))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a3)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2)
In Proofgold the corresponding term root is 799f7a... and proposition id is 2c046b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMT19iRdtVT39TEwjPkRGodbdZhE45fdCww)
ain a0 a2 → ain a0 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a1 a1) → (¬ (a0 = a2)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ ain (apow a2) a2) → (¬ (asing a1 = asing a2)) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (apow (asing a1) = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asm (apow a2) a2 = apow a2)) → ain a0 (apow (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a2) a4) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain a0 (asing a3)) → ain (asing a1) (aun (asing (asing a1)) a4) → adisjoint (apow (asing a1)) (asm a4 a2) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm a3 (asing a2) = a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) a4 → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 a4) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow a2) (asing a2)) → aal3 (asm (apow a2) (asing a1)) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is ba9521... and proposition id is 6fb98d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJJ1B4LSaiPjT86s1rAsdaRZwV8TnUMYAM)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → asubq a1 a2 → ain a0 (apow a2) → ain a0 (asm (apow a2) (asing a2)) → (¬ ain (asing a1) a2) → (¬ ain a1 (asm a3 (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → ain a0 (apow (asing (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a0 (asing a3)) → (¬ ain a2 (aun a1 (asing a3))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal5 (aun (asing (asing a2)) a4) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 a2 → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex5 (aun (asing (apow (asing a1))) a4) → (¬ ain (apow a2) a2)
In Proofgold the corresponding term root is d58499... and proposition id is 7a3e9f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMY2wbBRgixKCZYusiP8ypKWyJQicu3phwd)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ ain a1 (asing a2)) → (¬ (a1 = asing a2)) → (¬ ain a3 (asing a1)) → (¬ (a2 = asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (apow a2) a3) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain a1 (asing a3)) → ain a3 (asing a3) → (¬ ain (asing (asing a1)) a4) → ain a3 (asm a4 (asing a2)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal6 (aun a4 (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm a4 (asing a3)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex5 (aun a4 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → asubq a1 (apow (asing a1))
In Proofgold the corresponding term root is 3475c4... and proposition id is a5a32e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKXawqwGmx22mG2wX7kEuCuezkZhPayiqE)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (¬ (a0 = a2)) → (¬ (a0 = a3)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → asubq (asing a1) a2 → (¬ (a0 = asing a2)) → (¬ (a0 = asm a3 (asing a1))) → ain a2 (apow a2) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (asing a2) a2) → (¬ ain (apow a2) a2) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ (a0 = apow a2)) → (¬ (a1 = apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ (a3 = apow a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a0 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain (apow (asing a1)) a4) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (asing (asing a2)) a4)) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ ain a0 a0)
In Proofgold the corresponding term root is 152e01... and proposition id is fb4e56...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTRD7ABLGMcFD5sas5sJJ4je48BH9xQtYL)
ain a1 a2 → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → ain a1 (asing a1) → ain a1 (apow a2) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ ain (asing a1) a1) → ain (asing a1) (apow (asing a1)) → (¬ ain (asing a1) (asing a2)) → ain a0 (apow (asing a2)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asing a2)) → (¬ (asing a1 = a3)) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) a3) → (¬ (asing a2 = a3)) → (¬ (asing a2 = asm (apow a2) a2)) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a2) (apow (asing a2)) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain a1 (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal5 (aun a4 (asing (apow a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (apow a2) → aex5 (aun (apow a2) (asing (asing a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25))
In Proofgold the corresponding term root is a58fdb... and proposition id is ccd631...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYPAnmiycbbjrZfHZMZfiUmz1KQamEUQzT)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (¬ ain X24 a0)) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → asubq a1 a2 → (¬ ain (asing a1) a1) → ain a2 (asm a3 (asing a1)) → (¬ ain a3 (asing a1)) → (¬ (asing a1 = asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a1) (asm a4 a2)) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal3 a2) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (asm (apow a2) (asing a1)) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1))))
In Proofgold the corresponding term root is e3adfe... and proposition id is 32e58c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUyK69FMGDq4d4KNBVH5uGiw64JVSKixGe)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (¬ ain a3 a2) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ ain a1 (asing a2)) → (¬ asubq a2 (asing a2)) → (¬ ain a2 (apow (asing a2))) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ ain a2 (asing (asing a1))) → (¬ (a2 = asing (asing a1))) → (¬ (apow (asing a1) = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain a0 (apow (apow (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → (¬ ain a1 (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain a0 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a1) = a1) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a4 (asing a3)) a3 → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 a4 → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq a4 (aun a4 (asing (apow a2)))
In Proofgold the corresponding term root is 96e7ac... and proposition id is 2d2640...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNkPgCqqebEj7FxJjWDVMM6NsL55FYHeM9)
ain a1 a2 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (¬ ain a2 a2) → (¬ ain a3 a3) → asubq a3 a4 → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (¬ (a1 = asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ (a0 = apow (asing a1))) → asubq a1 (asm a3 (asing a1)) → (¬ ain a3 (asing a1)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (asing a2 = a3)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a3 (asm a4 (asing a1)) → ain a2 (asm a4 (apow (asing a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal2 a1) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal5 (aun a3 (asing (apow a2)))) → aal2 (asm a4 a2) → aal3 (asm a4 (asing a1)) → aal4 (apow a2) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (asm a3 (asing a2)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex3 (asm a4 (asing a0)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1))
In Proofgold the corresponding term root is 395d0b... and proposition id is 5ee47d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGTXECN5gK46w8h2qAWP4j6sgF6nqW5PPt)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a0 (apow a2) → (¬ ain a0 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = asing (asing a1))) → (¬ (a0 = asing a2)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asing a2) (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (asing a2))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (asm a4 (asing a3)) a3 → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal2 (asm (apow a2) a2) → aal3 (aun (asing a2) (apow (asing a2))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (asm a4 a2) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25))
In Proofgold the corresponding term root is f7aaf2... and proposition id is bdf737...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMaEvZBEvZs2dn9Jc8CtAzPvz6nivw1uHkL)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a0 a4 → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (¬ (a0 = a3)) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a0 (asm (apow a2) (asing a2)) → ain (asing a1) (asm (apow a2) a2) → asubq a1 (apow (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ (a2 = apow (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asm a3 (asing a1)) a4) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a1)) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 a3 → aal4 (apow a2) → aal5 (aun (asing (asing a1)) a4) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm a4 (asing a3)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is 39b109... and proposition id is 007483...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQFscpYezmjE2W4z7KLRbLxPZJb3pKbF29)
(∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a0 a1 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ (a0 = a1)) → ain a0 (asm (apow a2) (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain (asing (asing a1)) a3) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a0 (asm a4 (asing a2)) → (¬ ain (apow (asing a1)) a4) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a4 (aun (asing (asing a1)) a4) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (asm a2 (asing a0) = asing a1) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 (asm (apow a2) (asing a2)) → aex3 (asm (apow a2) (asing a2)) → aex2 (asm a3 (asing a2)) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → aex3 (aun a2 (asing (apow a2)))
In Proofgold the corresponding term root is 3eb6c9... and proposition id is 5eedda...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNQv2S81bd2Rx39DZXUWoZNbceReTKS7hs)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a2 a1) → (¬ (a1 = a3)) → (¬ asubq a1 a0) → ain (asing a1) (asing (asing a1)) → (¬ ain a0 (asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → ain a2 (asm (apow a2) (asing a1)) → (¬ (a1 = apow (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → asubq a3 (apow a2) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a1 (apow (apow (asing a1))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain a0 (aun (asing (asing a2)) a4) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a3 (aun a3 (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (asm a4 (asing a3) = a3) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) a4 → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 a2 → aal3 (aun (asing a1) (apow (asing a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (asing (asing a1)) a4) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm a4 a2) → aex2 (aun a1 (asing (apow a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ ain a2 a2)
In Proofgold the corresponding term root is 57fb6e... and proposition id is 05589d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYeVLeSecvCzjbpdk2wwuwpBjg24Yw7zRk)
ain a0 a4 → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (¬ ain a3 a2) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain (asing a1) (apow a2) → (¬ ain a2 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain a0 (apow (apow (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a2 (asing a3)) → ain a1 (asm a4 (apow (asing a2))) → ain a0 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asing a2) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq a1 (asm a2 (asing a1)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (asm (apow a2) (asing (asing a1)) = a3) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) a4 → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow a2) (asing a2)) → aal5 (aun (apow a2) (asing (apow a2))) → aex3 (aun a2 (asing (apow a2))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a1) (asm (apow a2) a2)
In Proofgold the corresponding term root is 22320f... and proposition id is c3633f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLqtu3TzvWWrVfb15MoU7pvE6UrWrYaE8F)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → asubq a1 (asing a0) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, aex2 (apow (asing X24))) → ain (asing a1) (asm (apow a2) a2) → (¬ ain a1 (apow (asing a2))) → ain a2 (asm (apow a2) a2) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ asubq (asing a2) a2) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a2) a4) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a1 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asing a1) (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aex2 (apow (asing a1)) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (apow (asing a2)) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (asm a4 (asing a1)) → aex3 (asm (apow a2) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ (a1 = asm (apow a2) a2))
In Proofgold the corresponding term root is e4b03a... and proposition id is 594b0f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKS9kGp5WzMFSWgaRmTqaKpM4FRttQD3aa)
(∀X24 : set, (¬ ain X24 a0)) → ain a0 a2 → ain a1 a2 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a0 a0) → (¬ ain a3 a3) → (¬ (a0 = asing (asing a1))) → (¬ ain (asing a1) (asing a2)) → (¬ (a1 = asing a2)) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain a3 (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (asing a2 = apow a2)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a1)) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → aal3 (asm (apow a2) (asing a2)) → aal3 a3 → aex2 (asm a3 (asing a2)) → aex3 (asm (apow a2) (asing a1)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal3 (apow (asing (asing a1))))
In Proofgold the corresponding term root is da64f7... and proposition id is f68ba7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbz9LUrZWGQdiHiYC5TWE3FvNW1SZar8sH)
(∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ asubq a2 a1) → (¬ (a2 = a3)) → (¬ ain a0 (asing (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (asing a1)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ asubq a3 (asing a2)) → (¬ (a0 = apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a2 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing a2)) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 a3 → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) a4)
In Proofgold the corresponding term root is 793b95... and proposition id is 2dac08...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdEnw6NLBRzJDwtKRtFU9y3pewqvFyM3GE)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain a0 (asing a2)) → ain a0 (asm a3 (asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ ain (asing a1) a1) → (¬ (a0 = asm a3 (asing a1))) → asubq a1 (asm a3 (asing a1)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a0 (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a2 (aun a1 (asing a3))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a3 (asm a4 (asing a1)) → ain a3 (asm a4 (apow (asing a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 a2) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal3 (aun (asing a2) (apow (asing a2))) → aal3 (aun a2 (asing (apow a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun a4 (asing (apow a2))) → aex2 (asm a3 (asing a1)) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (aun a2 (asing (apow a2))) → aex3 (asm (apow a2) (asing (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is d42204... and proposition id is 64fba2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMafo5fMvccnPMApkQMAKt8rBAaXyGJtYty)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a1 a2 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a0 (apow (asing a1)) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (asm (apow a2) a2) a3) → (¬ (asm a3 (asing a1) = a3)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a3 (asing a3) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a0 (aun a3 (asing (apow a2))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq a1 (asm a2 (asing a1)) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (asm (apow a2) a2) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm a3 (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex4 (apow a2) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))
In Proofgold the corresponding term root is fb9725... and proposition id is 04bbdf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZRPCoPASG4wFZbN1cnfeFJAwP8sMNtP6Q)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → asubq a1 (asing a0) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (¬ ain a2 a2) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a0 (apow a2) → (¬ ain a0 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a0 (asing (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (asing a1) (asing a1)) → (¬ (asing a1 = asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) a2) a3) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (asm a4 (asing a1)) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a3 (aun a1 (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain (asing a1) (asm a4 a2)) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a3 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing (asing a1))) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (asm a4 a2) → aex2 (asm a3 (asing a1)) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (¬ ain (asing a1) a0)
In Proofgold the corresponding term root is 310e7f... and proposition id is a5252a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFGyxQVjzqCCM4y3uWbvMnrPGBQdzDgZwe)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → ain (asing a1) (asing (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a1 (apow (asm a3 (asing a1))) → ain a3 (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain a0 (asing (apow a2))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun a4 (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1))
In Proofgold the corresponding term root is 696bf9... and proposition id is 206b43...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNHcLp8Ri9Yk6YocBQ2naUPnJCBAjNAywA)
ain a0 a2 → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ asubq a2 a0) → (¬ ain (asing a2) a1) → asubq a1 (apow (asing a1)) → (¬ ain a2 (apow (asing a1))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (apow a2) a3) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a1 (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm a3 (asing a2) = a2) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 a1) → (¬ aal3 (apow (asing a1))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (aun (asing a2) (apow (asing a2))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex4 a4 → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 46009e... and proposition id is 852483...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTD7qGSWLabMwT9BVYFYMxjW1k2U1iUcSg)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a1 a3 → ain a1 a4 → ain a3 a4 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a3 a3) → (¬ (a2 = a3)) → (¬ ain (asing a1) a0) → ain (asing a1) (asing (asing a1)) → (¬ (a0 = asm (apow a2) a2)) → ain a2 (asm a3 (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain (apow a2) (apow a2)) → (¬ (a0 = apow a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a0 (apow (asing (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (apow (asing a1))) a4) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aun a1 (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex5 (aun (apow a2) (asing (asing a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1))
In Proofgold the corresponding term root is bd53d7... and proposition id is 35d57f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdD5anGu9EQZGCqEZhonEjQpswLDmgqAso)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → asubq a2 a3 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (asing a1) → ain a1 (asm (apow a2) (asing a2)) → ain a2 (asm a3 (asing a1)) → (¬ ain (asing (asing a1)) a2) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ (a2 = apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (asing a2 = a3)) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a2) (asing (asing a2)) → ain a1 (asm a4 (apow (asing a2))) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 a3 → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24))))
In Proofgold the corresponding term root is 9396d1... and proposition id is 09e535...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMnFVAhDM5UHJuLX9ptAJkMX38vsaCHLme)
(∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → asubq a1 a2 → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = apow a2)) → (¬ ain (apow a2) (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ ain (asing a2) a3) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain a1 (asing a3)) → (¬ ain (asing a2) (aun a1 (asing a3))) → (¬ ain (asing (asing a1)) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a1 (asm a4 (apow (asing a2))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a2) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm a4 (asing a3)) a3 → asubq a2 (apow (asm a3 (asing a1))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm a4 a2) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (asm a4 (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm a4 (asing a0)) → aex3 (asm a4 (asing a3)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))))
In Proofgold the corresponding term root is 46e0b8... and proposition id is 6c26d2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMS4QwA997HGBCdWYx9mNo58ghoncRMYbWD)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a1 a2 → ain a0 a3 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a0 a0) → asubq a1 a2 → (¬ asubq a2 a1) → asubq a3 a4 → (¬ (a1 = a2)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a1 (asing a1) → asubq a1 (apow (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ ain (asing a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ (a2 = apow a2)) → (¬ (apow (asing a1) = a3)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ (a3 = apow a2)) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain (asing a2) (asing (apow a2))) → (¬ ain a3 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (asm a4 (asing a3) = a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (¬ aal2 a1) → (¬ aal2 (asing a3)) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 a3 → aal3 (aun (asing a2) (apow (asing a2))) → aal3 (aun a2 (asing (apow a2))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 a4 → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun a1 (asing a3)) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (asm a4 (asing a2)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2))
In Proofgold the corresponding term root is c8e4a9... and proposition id is f37b90...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWUVgk3V5okYRBoM3AvZ5CjUCFFKHKJCy3)
(∀X24 : set, (¬ ain X24 X24)) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a2 a1) → (¬ (a0 = a2)) → ain a2 (asing a2) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain a1 (apow (asing a2))) → (¬ (asing a1 = a2)) → (¬ asubq a2 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a2))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → (¬ (a3 = apow a2)) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a1 (aun (asing (asing a2)) a4) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 (asing (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm a3 (asing a2)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is 70f6b1... and proposition id is 822507...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbaEd5SpXQrdyvbupkrCKzR2YDSF67jLp6)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (¬ ain a3 a2) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (¬ (a0 = apow (asing a1))) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a1))) → (¬ ain a3 (asing a1)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing (asing a1)) a3) → (¬ ain a3 (asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (asing a2 = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a2 (asing a3)) → ain a0 (asm a4 (asing a2)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a0 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a2 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (apow (asing a2)) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (apow a2) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asing (asing a2)) a4) → aex5 (aun a4 (asing (apow a2))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24))))
In Proofgold the corresponding term root is a4f4fa... and proposition id is 918add...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGZiHvX2QBFjnX3LVSYczVxSu5biExq7x8)
(∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain (asing a1) (apow (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain a3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = apow a2)) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain (asing a2) (aun a1 (asing a3))) → (¬ ain (asing (asing a1)) a4) → (¬ ain (asm a3 (asing a1)) a4) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a1 (asm a2 (asing a1)) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal4 a4 → aex3 (asm a4 (asing a2)) → aex3 (asm a4 (asing a0)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aun a1 (asing a3))
In Proofgold the corresponding term root is d6350c... and proposition id is 84912e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdYdpBg7bkJbHJAJ9XZBeSqaMmHTvMcvvH)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (¬ ain a0 (asing a2)) → ain a0 (asm (apow a2) (asing a2)) → (¬ (a0 = asing a1)) → ain a2 (asm (apow a2) a2) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → ain a1 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a1 (aun (asing a1) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (asing (asing a2)) a4) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asing a1) (asm a2 (asing a0)) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (asm a4 a2)) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex2 (asm a3 (asing a1)) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2)))
In Proofgold the corresponding term root is de1852... and proposition id is ee8b01...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcGY7Euxkmac6TsKV15D487FGBnir2uaWW)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → ain a0 a3 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a1 a0) → (¬ asubq a3 a2) → ain a2 (asing a2) → (¬ ain (asing a1) a2) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a1 (asing (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) a3) → (¬ (asing a2 = asm (apow a2) a2)) → ain a0 (apow (asing (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain a3 (asing (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain (asing (asing a1)) a4) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a1 (aun a2 (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq a3 (asm a4 (asing a3)) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aun a1 (asing (apow a2))) → aex3 a3 → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex5 (aun (asing (asing (asing a1))) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ asubq a2 (asing a1))
In Proofgold the corresponding term root is edb2f3... and proposition id is bd57be...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWF3na4MGUzRKXUyzvEW39nCCK3sdhk1P8)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a0 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → asubq a3 a4 → (¬ asubq a1 (asing a2)) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (asing a1)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asm a3 (asing a1)) a2) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a3 (aun (asing a1) (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → (¬ ain a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal6 (aun (asing (asing a2)) a4)) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (apow a2) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is 4733ce... and proposition id is 642ba4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVQbAP1x9fpX7zc7fEFQenpMLKULFh7Mbb)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a2 a1) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (¬ ain a1 (apow (asing a1))) → (¬ ain (asing a1) a2) → (¬ asubq a1 (asing a2)) → (¬ ain a1 (asing a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a1))) → (¬ asubq a2 (asing a2)) → (¬ ain a3 (asing a1)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asm (apow a2) a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq a3 (apow a2) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (apow a2) a4) → (¬ ain a0 (asm a4 a2)) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (¬ aal3 a2) → (¬ aal3 (asm a4 a2)) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow a2) (asing a2)) → aal3 (aun (asing a1) (apow (asing a2))) → aex2 (aun a1 (asing (apow a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun a3 (asing (apow (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is dfa117... and proposition id is c8c625...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUrBfRq3VrhS8zDkH3ijC1Gk34jbVW81NK)
(∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → ain a1 (apow a2) → ain (asing a1) (apow a2) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (apow a2) a2) → (¬ asubq (asing a2) a2) → (¬ asubq a2 (asm a3 (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (a2 = asing a2)) → (¬ ain (apow (asing a1)) a3) → (¬ ain a3 (asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) a3) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a2 (aun a1 (asing a3))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (apow (asing a2))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ ain a2 a0)
In Proofgold the corresponding term root is 0f9bcb... and proposition id is 5cba07...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHAJtz6Wwg79Nj3BTuzyDF8WAvemHnb48s)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (asing a1) → ain a2 (asing a2) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ (a0 = apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a0 (asm a4 (asing a2)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal3 a3 → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing (asing a1))) a4) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → ain (apow a2) (aun a2 (asing (apow a2)))
In Proofgold the corresponding term root is 4e1d2d... and proposition id is 94ff72...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMEpC4goMTLzXZJ5FHEqsk6jkL9jDektkg3)
(∀X24 : set, (¬ ain X24 a0)) → ain a0 a1 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (¬ ain a2 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ ain (asing a1) a0) → ain a0 (asm a3 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → asubq (asing a1) a2 → (¬ ain (asing a1) a2) → (¬ (a0 = asm a3 (asing a1))) → (¬ (a1 = asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a1 (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (a2 = apow a2)) → (¬ ain (apow a2) a3) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a0 (asing a3)) → ain a0 (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 a2 → aal3 (aun (asing a2) (apow (asing a2))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a1)) → aex3 (asm a4 (asing a1)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1))
In Proofgold the corresponding term root is 2a341a... and proposition id is e20587...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSgWX4LHCs5v5ascGvjzqqfwJQnJi2eoXS)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a0 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a2 a1) → (¬ (a0 = a1)) → ain a0 (apow a2) → ain a0 (asm a3 (asing a1)) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a0 (asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (apow a2) (asing a2)) → asubq (asm a3 (asing a1)) (apow a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (apow (asing a1) = a3)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (a3 = apow a2)) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (apow (asing a2)) → (¬ ain (asing a2) (asing a3)) → ain a3 (aun a1 (asing a3)) → ain a3 (asm a4 (asing a2)) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun a1 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain a2 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (asm (apow a2) (asing a2)) → aal4 (aun a3 (asing (apow (asing a1)))) → aal4 a4 → aal4 (aun a3 (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a2)) → aex5 (aun (asing (apow (asing a1))) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 a3
In Proofgold the corresponding term root is 8f0079... and proposition id is c77d56...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMY6RHde6viowrVdiyYRbv84V6ZaoKzZtxd)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → asubq (asing a0) a1 → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain a1 (asing a1) → ain a2 (asing a2) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ ain (apow a2) (asing a2)) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a2 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (¬ aal2 (asing a1)) → (¬ aal3 a2) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 a2 → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (asm (apow a2) (asing a2)) → aal4 (apow a2) → aex2 (aun a1 (asing a3)) → aex2 (asm a4 a2) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex5 (aun (asing (asing a1)) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain (asing (asing a1)) (apow (asing (asing a1)))
In Proofgold the corresponding term root is 5e236e... and proposition id is 24e8b1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMc5PMSaiMRERaUbdZAs3ZGjxWr9UwqXkcQ)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a2 a2) → (¬ ain a3 a3) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a0 (asm (apow a2) (asing a2)) → ain a1 (apow a2) → asubq a1 (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing (asing a1)) a3) → asubq a3 (apow a2) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asing a2 = apow a2)) → ain a0 (apow (asing (asing a1))) → ain a1 (apow (apow (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a3 (asing a3) → (¬ ain a2 (aun a1 (asing a3))) → ain a3 (asm a4 (asing a1)) → ain a2 (asm a4 (apow (asing a2))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (¬ aal4 a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (aun a1 (asing a3)) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (asm (apow a2) (asing a2)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))))
In Proofgold the corresponding term root is 9c5e87... and proposition id is b48e27...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPhhkCFgVs2iXYHYTLkphzohoJPk4m59ET)
(∀X24 : set, (¬ ain X24 X24)) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a1 a0) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a2 (asing a2) → (¬ asubq a1 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ (a0 = asing a2)) → ain (asing a1) (apow (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ (a1 = asing a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq a2 (asm a3 (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (asing a2 = a3)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain a0 (asing a3)) → (¬ ain a1 (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asing a1) (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm a3 (asing a2)) a2 → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal4 a3) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (aun a1 (asing a3)) → aex3 (asm a4 (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is 7d9534... and proposition id is e1e9c7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFQLbKCGHhzQE3RAt76nY1qfMCJ2perSF3)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → ain a0 a1 → (∀X24 : set, asubq a0 X24) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (¬ ain a1 a0) → (¬ ain a3 a2) → (¬ (a1 = a3)) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (asing a1) → (¬ ain a0 (asing a2)) → (¬ ain (asing a1) a2) → asubq a1 (asm a3 (asing a1)) → (¬ (a1 = asm a3 (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain a2 (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) a3) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asing a2 = apow a2)) → ain a0 (apow (apow (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a0 (asm a4 (asing a2)) → (¬ ain a0 (asm a4 a2)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (asing (asing a1)) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (apow a2) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → ain a0 (aun (apow (asing a2)) (asing a3))
In Proofgold the corresponding term root is b0e7b1... and proposition id is 5a5445...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFYVLi8CJngoT6QChM6YcuvgWjtoHrSfoa)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → ain a0 a1 → ain a0 a2 → ain a2 a3 → ain a1 a4 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (¬ (a0 = a1)) → ain a0 (apow a2) → ain a1 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (asm (apow a2) a2) → (¬ ain a1 (apow (asing a2))) → (¬ ain (asing a2) a1) → (¬ ain a3 (asing a1)) → (¬ (asing a1 = a3)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (asm a3 (asing a1))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asing a1) (aun (asing (asing a1)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a0 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex2 (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 a4 → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2)))
In Proofgold the corresponding term root is f25c95... and proposition id is ca2831...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdoq8jMQGyQessw39ogWjVNimvNXNpZZtK)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a1 a0) → (¬ (a0 = a2)) → (¬ asubq a2 a0) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a2 (asm (apow a2) a2) → (¬ ain (asing a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (asm a4 (asing a1)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → aal3 a3 → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (¬ aal4 (aun a2 (asing (apow a2))))
In Proofgold the corresponding term root is 91b3e4... and proposition id is 460f7c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYz8UctHw8ZdGDvYYdeYRM2FeE29PMJhvj)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a2 a3 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a2 a0) → (¬ asubq a2 a1) → (¬ asubq a4 a3) → ain a1 (asm (apow a2) (apow (asing a2))) → asubq a1 (apow (asing a1)) → (¬ (a1 = asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (a0 = apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asing (asing a1)) a4) → ain a3 (asm a4 a2) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a4 (aun a4 (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (asm (apow a2) (asing (asing a1)) = a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (¬ aal2 (asing a3)) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → aal2 a2 → aal4 (aun a3 (asing (apow (asing a1)))) → aal4 a4 → aex2 (aun (asing a3) (asing (apow a2))) → aex3 a3 → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1)))
In Proofgold the corresponding term root is a48a77... and proposition id is 937d1b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQH5qid4H5uLAPzyuNWSJMPfnJkgBXWKqc)
ain a3 a4 → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → asubq a3 a4 → (¬ (a2 = a3)) → ain a1 (asing a1) → (¬ (a0 = asing a1)) → (¬ (a0 = asm (apow a2) a2)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asing a2) a2) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (asm a3 (asing a1) = a3)) → (¬ (asing a2 = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a3 (asing a3) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (apow (asing a1)) → aex3 (asm (apow a2) (asing a2)) → aex2 (asm a3 (asing a2)) → aex3 (asm (apow a2) (asing a1)) → aex5 (aun (asing (apow (asing a1))) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))
In Proofgold the corresponding term root is ca2c5c... and proposition id is b31819...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXZftYxV5jdT8CUCUcBeVj1BQ9wr3rdqj2)
(∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → asubq a2 a3 → (¬ ain a1 (apow (asing a1))) → ain (asing a1) (apow a2) → asubq a1 (apow (asing a1)) → (¬ (asing a1 = asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ (apow (asing a1) = a3)) → (¬ (asing a2 = a3)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain a2 (asm a4 (asing a1)) → (¬ ain a1 (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal5 (apow a2)) → (¬ aal5 a4) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm (apow a2) a2) → aal5 (aun (apow a2) (asing (asing a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm a4 (asing a0)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → ain (asing a2) (aun a3 (asing (asing a2)))
In Proofgold the corresponding term root is 35f86d... and proposition id is 1eb88b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSYRxGh7FUeC1bNrzrq8jmcmVpVTEYT5n8)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a3 a2) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ (asing a1 = a2)) → (¬ ain (asing a1) (asing a1)) → (¬ ain a1 (asing (asing a1))) → (¬ (asing (asing a1) = a3)) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (apow a2) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal2 (asm (apow a2) a2) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing a2)) a4) → aex2 (asm (apow a2) a2) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm a4 (asing a0)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25))
In Proofgold the corresponding term root is d1fc90... and proposition id is 9bd08f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUtPBhLuRzBMx7ZLs1cEXzeauNQ7P4XKj2)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a1 a2 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ (a0 = a1)) → (¬ (a2 = a3)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asing a2)) → (¬ asubq a2 (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = a3)) → (¬ (asing a1 = asing (asing a1))) → (¬ ain (asing a1) a3) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) a3) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a2 (asm a4 a2) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing (asing a1))) → (¬ aal3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal4 (apow a2) → aal4 a4 → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex3 (asm a4 (asing a0)) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1)))))
In Proofgold the corresponding term root is f3803e... and proposition id is efd3fe...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHQL9pkqPkjmpqZpH71QtJEPuPgfYV1JRA)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → asubq a1 (asing a0) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain (asing a1) (apow a2) → (¬ (a1 = asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (apow a2) a2) → (¬ asubq (asing a2) a2) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a0 (apow (asing (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (asing a3)) → ain a3 (asm a4 (asing a2)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain a0 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → asubq a4 (aun a4 (asing (apow a2))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aun (asing a3) (asing (apow a2))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (apow (asing a1))) a4) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))
In Proofgold the corresponding term root is afd24c... and proposition id is 59996f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFn3FWnMQ4JSNViDvXeXKpHBGAV2AAkQHq)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ asubq a3 a2) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → ain (asing a1) (asing (asing a1)) → ain a0 (apow a2) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ asubq a2 (asing a2)) → (¬ (a1 = asing a2)) → (¬ ain (apow a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ ain a0 (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain a2 (asm a4 (asing a1)) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain (apow (asing a1)) a4) → (¬ ain (apow a2) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm a4 (asing a3)) a3 → asubq (asm a3 (asing a1)) a4 → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aun a2 (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a3)) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25))))
In Proofgold the corresponding term root is c2fafa... and proposition id is 97b2ad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGubrCSaDoqJ1otDuVp2pakbqaYYyJHdJq)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (¬ ain a2 a0) → (¬ ain a3 a3) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a0 (apow a2) → ain a2 (asing a2) → ain (asing a1) (apow a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a2 (apow a2) → ain a2 (asm (apow a2) a2) → ain a2 (asm (apow a2) (asing a1)) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ (a2 = apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a2 = asm (apow a2) a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a1) (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (asm a4 (asing a3) = a3) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (apow a2)) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (aun a2 (asing (apow a2))) → aex2 (asm a3 (asing a1)) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm a4 a2) → aex5 (aun (asing (asing (asing a1))) a4) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ (a0 = asing a1))
In Proofgold the corresponding term root is d88588... and proposition id is 92cf5f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcHYzd9PcxkddMMCtyrCCam3sQPfzLT1N7)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ (a0 = a2)) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → ain (asing a1) (asing (asing a1)) → ain a0 (apow (asing a1)) → (¬ (a0 = asm a3 (asing a1))) → (¬ (a1 = asing a2)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain (asing (asing a1)) a3) → (¬ ain (apow a2) a3) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain a1 (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a3 (asing a2) = a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm a3 (asing a1)) a4 → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal2 a1) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal3 a3 → aal5 (aun (asing (apow (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex4 (aun a3 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))
In Proofgold the corresponding term root is 21ffe1... and proposition id is fc3510...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZ3aQSu4bjqGgpHctaoaJXEhZi6mbiq2AM)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (¬ ain (asing a1) a0) → ain a0 (apow (asing a1)) → ain a0 (apow a2) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) (asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asing (asing a1)) a2) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (apow a2) (apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a3 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a4 (aun a4 (asing (apow a2))) → (asm a2 (asing a0) = asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) a4 → asubq (asm (apow a2) (apow (asing a1))) a4 → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 a1) → (¬ aal2 (asing a3)) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (asm a3 (asing a1)) → aex2 (asm a4 a2) → aex3 (asm (apow a2) (asing a2)) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))))
In Proofgold the corresponding term root is 64896e... and proposition id is 4e2e0d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdY8hYec4xTxvzDSBzXiTMC7qpc2rQn2ck)
(∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → asubq a3 a4 → (¬ asubq a4 a3) → (¬ ain a1 (apow (asing a2))) → (¬ ain (asing a1) (asing a2)) → (¬ ain a2 (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ asubq a2 (asing a2)) → (¬ (a1 = asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (apow a2) (apow a2)) → (¬ ain (asing a2) a3) → (¬ (asm a3 (asing a1) = a3)) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 a4) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (aun (asing a2) (apow (asing a2))) → aal5 (aun (asing (asing a2)) a4) → aex2 a2 → aex3 a3 → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → ain (asing a1) (asm (apow a2) (asing a2))
In Proofgold the corresponding term root is ffd037... and proposition id is 5acaf9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdNciLyfityckRuWh15ufDvNXi23JorvTq)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a1 (apow (asing a1))) → (¬ (a0 = asing (asing a1))) → ain a2 (asm a3 (asing a1)) → ain a0 (apow (asing a2)) → (¬ (a1 = apow (asing a1))) → (¬ asubq (asm (apow a2) a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asing a2) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ ain (asing (asing a1)) a3) → (¬ (a1 = apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a1)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun (asing (asing a1)) a4) → aex2 (asm a3 (asing a1)) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asing (asing a1)) a4) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1)))
In Proofgold the corresponding term root is d9452e... and proposition id is f73f59...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZwCGjVgYfqfEq4noma3RnfzvywwNZgqud)
ain a1 a3 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a2 a1) → (¬ (a0 = a1)) → ain a1 (asing a1) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) a1) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a2 (asing a1)) → ain a2 (asm a3 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a1))) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain a2 (apow (asing a2))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq a3 (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → ain a1 (asm a4 (asing a2)) → (¬ ain (asm a3 (asing a1)) a4) → ain a3 (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq a1 (asm a2 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal3 (asm (apow a2) (apow (asing a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex4 a4 → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ ain (asing a2) (asm (apow a2) (asing a1)))
In Proofgold the corresponding term root is 1fca9a... and proposition id is d2c902...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFgkFLSVWJrZwszQEUoAvC8KoGyd2km9DK)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a0 a2 → ain a0 a3 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → (¬ ain a3 a2) → (¬ ain a3 a3) → (¬ asubq a2 a1) → ain a0 (asm a3 (asing a1)) → (¬ asubq a2 (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (apow (asing a1)) a3) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain (apow a2) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a0 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm (asm a4 a2) (asing a3) = asing a2) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal4 (apow a2) → aal4 (aun a3 (asing (asing a2))) → aex2 (asm (apow a2) a2) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is b2cb10... and proposition id is 828e25...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKd5WLuNSDeDnj8nmLod3YxdmEPNMZPUr5)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → asubq a2 a3 → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a0 (asm (apow a2) (asing a2)) → ain (asing a1) (apow a2) → (¬ (a1 = asing a1)) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing a2) a3) → (¬ (a3 = apow a2)) → ain a0 (apow (asing (asing a1))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (asm a4 (asing a1)) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) a4) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a0 (asm a4 a2)) → ain a2 (asm a4 a2) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a3 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 a2) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal5 (aun a4 (asing (apow a2))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex5 (aun (asing (apow (asing a1))) a4) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ (a1 = apow a2))
In Proofgold the corresponding term root is 763187... and proposition id is 4ee984...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNS9BShEGU76nNrXpRDncRk42bvKi8gSqt)
(∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a0 a0) → (¬ ain a2 a0) → (¬ (a1 = a3)) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain (asing a1) (asing (asing a1)) → ain a1 (aun (asing a1) (asing (asing a1))) → ain (asing a1) (asm (apow a2) a2) → (¬ ain a1 (asing (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = a3)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain a2 (aun (asing (asing a2)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun a4 (asing (apow a2))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a1 (asm a2 (asing a1)) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 (asm a4 a2) → aal5 (aun (asing (asing a1)) a4) → aex2 (aun (asing a1) (asing (asing a1))) → aex4 (apow a2) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex5 (aun (asing (asing a2)) a4) → aex5 (aun (apow a2) (asing (apow a2))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ (a1 = apow a2))
In Proofgold the corresponding term root is 505d55... and proposition id is a674f4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSn7W8wri52wiE1pWca7gxt4THXwRUApFf)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (¬ ain a0 (asing a1)) → asubq (asing a1) a2 → (¬ (a0 = asing (asing a1))) → (¬ ain (asing a1) (asing a2)) → ain a2 (apow a2) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a3) → (¬ (asm a3 (asing a1) = a3)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (asm a2 (asing a0)) (asing a1) → asubq a1 (asm a2 (asing a1)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal2 (apow (asing (asing a1))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aun a1 (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex5 (aun (asing (asing a2)) a4) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ ain (asing a2) (asm (apow a2) a2))
In Proofgold the corresponding term root is ee672f... and proposition id is 4d4110...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVX6xG72g8j1ZKWKHDLmgGLuLt7oTiGERc)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ ain a2 a1) → (¬ ain a2 a2) → (¬ ain (asing a1) a0) → ain a2 (asing a2) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a1 (asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ asubq (apow a2) (asing a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (a3 = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a0 (apow (apow (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain (asing a1) (asm a4 a2)) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a4 (aun (asing (asing a1)) a4) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm a2 (asing a1)) a1 → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 a4 → aal5 (aun (apow a2) (asing (asing a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25))
In Proofgold the corresponding term root is 108772... and proposition id is e89c85...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGNfLoCgXqZ9EmGzHNf2sHAqN1RxB4Vzyo)
ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a3 a2) → (¬ (a0 = a1)) → ain a0 (apow a2) → (¬ ain (asing a1) a2) → (¬ (a0 = asing a2)) → (¬ ain (asing a2) a1) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (asing a2) a4) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a0 (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm a4 (asing a1)) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun a4 (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (apow (asing (asing a1))) → aal3 (asm (apow a2) (asing a1)) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (aun (asing a2) (asing (asing a2))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → ain (asing a1) (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is 77ba9f... and proposition id is 8d9174...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFqoD37YJCay7ExgRkEYYYqpxXXAD1bDMN)
(∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (¬ ain a2 a0) → ain a0 (apow a2) → (¬ asubq a1 (asing a1)) → (¬ ain (asing a1) a2) → (¬ (a1 = asm a3 (asing a1))) → (¬ asubq (asing a1) (asing a2)) → (¬ ain a1 (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = asing (asing a1))) → (¬ (a2 = asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (asing (asing a1)) a3) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain a1 (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm a3 (asing a2) = a2) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal2 (asing a3)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun (asing (asing a1)) a4) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (apow a2) a2) → aex2 (aun a1 (asing a3)) → aex2 (aun a1 (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24))))
In Proofgold the corresponding term root is 09fda0... and proposition id is 3fee9a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRw29v8vLQ6UwZizfR2prbDjJsQqSCf3ZJ)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (¬ ain a1 a1) → (¬ ain a3 a2) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a2 (asing a2) → (¬ ain a0 (asing a1)) → ain (asing a1) (apow a2) → (¬ ain a2 (apow (asing a2))) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = asing (asing a1))) → (¬ ain a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a1 (asm a4 (asing a2)) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing a2) (apow (asing a2))) → aal3 (asm a4 (asing a1)) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 a2 → aex2 (apow (asing a1)) → aex2 (asm a3 (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)))
In Proofgold the corresponding term root is ec15b7... and proposition id is 17b854...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSy7mLSskQ3BgBZrJaqMXK1cwGx3o92SsQ)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ (a0 = a3)) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain a1 (apow (asing a1))) → ain (asing a1) (apow a2) → (¬ asubq a1 (asing a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → asubq a1 (asm a3 (asing a1)) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asing a1) a3) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (apow a2) (asing a2)) → (¬ (asing a2 = apow a2)) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a3 (aun a1 (asing a3)) → ain a1 (asm a4 (apow (asing a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (asm a4 (asing a1)) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ ain (apow (asing a1)) a3)
In Proofgold the corresponding term root is 7ddc42... and proposition id is f1b05b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTmgWMuHyaYp4MY6JrXo5Rqeho9KTypndQ)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a3 a3) → (¬ asubq a3 a2) → (¬ (a0 = a1)) → (¬ asubq a1 a0) → (¬ asubq a2 a0) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain a2 (asing a1)) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asing a2)) → (¬ ain (asing a1) a3) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (apow a2) (apow a2)) → (¬ (asing a2 = a3)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a1 (aun a3 (asing (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (asing a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun a2 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → asubq a1 (asm a2 (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (asm a4 (asing a3) = a3) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (asm (apow a2) (asing a1)) → aex2 (aun a1 (asing a3)) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (apow a2))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, (¬ aal2 (asing X24)))
In Proofgold the corresponding term root is db5206... and proposition id is c3e277...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbLMELAy6r6Dw3Qt1y8oujfzavgdZtxoP9)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → asubq a3 a4 → (¬ ain (asing a1) a2) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a1 (asing (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ (a1 = apow (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (asing a1 = apow a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (a0 = apow a2)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 a2) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → (asm a2 (asing a1) = a1) → asubq a2 (asm a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (asm a4 (asing a2)) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex4 (apow a2) → aex3 (asm a4 (asing a0)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (apow a2) (asing (asing a2)))
In Proofgold the corresponding term root is de4a71... and proposition id is 53b0db...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMSQLGWT4DHgrPf2uH6fS2UijTtGT5BEam)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (¬ ain a3 a3) → asubq a2 a3 → (¬ (a0 = a1)) → ain (asing a1) (asing (asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → asubq a1 (apow (asing a1)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (asm a3 (asing a1) = a3)) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asing a1) (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asing (asing a2)) a4) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex5 (aun (asing (asing a1)) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2)))
In Proofgold the corresponding term root is 0f1f28... and proposition id is 9e60be...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbPw1fQhmgrjyoaMxbxLLZ3wrvLy4UXyq4)
ain a1 a4 → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a2) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (¬ ain (asing a1) a1) → ain a2 (asm a3 (asing a1)) → (¬ ain a2 (apow (asing a2))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asm a3 (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ ain (apow a2) a3) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain a2 (asing a3)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 a1) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 (asm (apow a2) a2) → aal4 (apow a2) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (asm (apow a2) a2) → aex2 (aun a1 (asing a3)) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (asm (asm a4 (asing a2)) (asing a3) = a2)
In Proofgold the corresponding term root is 778f89... and proposition id is a6b67e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRERUF53CXGXTATEZCExLAimCJQYLDwMeC)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ asubq a1 (asing a2)) → (¬ ain a2 (apow (asing a1))) → (¬ ain a1 (asing (asing a1))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a2 (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asing a2 = a3)) → ain a0 (apow (asing (asing a1))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (asing a3)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a2 (asm a4 (apow (asing a2))) → ain a1 (aun (asing (asing a2)) a4) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (asm a4 (asing a3) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (asm a4 (asing a2)) → aal3 (asm a4 (asing a1)) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm (apow a2) (apow (asing a1))) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (apow a2) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is efdb1c... and proposition id is b26ec4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVvkKjS2vpWKdiby8jpbNWP73t2wy7KavF)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asing a2)) → (¬ ain (asing a1) (asing a2)) → ain a2 (apow a2) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ (a1 = apow a2)) → (¬ (asing a2 = apow a2)) → (¬ (a3 = apow a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (apow a2) (asing (apow a2)) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a1 (asm a2 (asing a1)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal3 (asm a3 (asing a1))) → aal2 (asm a4 a2) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun a3 (asing (asing a2))) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (apow a2) (asing (asing a1))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1)))
In Proofgold the corresponding term root is c191ca... and proposition id is deead5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMX3f9LE3VZJ9KiZqnjNkWR8teinok45RaK)
(∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a2 a1) → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a1 (apow a2) → (¬ ain a1 (apow (asing a2))) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (a0 = asing (asing a1))) → (¬ (a1 = asm a3 (asing a1))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ (a3 = apow a2)) → ain a0 (apow (apow (asing a1))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a1 (asm a4 (asing a2)) → ain a3 (asm a4 (asing a2)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a2 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a2 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq a1 (asm a2 (asing a1)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (asing (asing a1))) a3 → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 a2 → aal3 (asm (apow a2) (asing a1)) → aal4 (aun a3 (asing (apow (asing a1)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm a3 (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm a4 (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → ain a1 a2
In Proofgold the corresponding term root is dedbee... and proposition id is c02d33...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcj4sTGhs7DM5wwRHhjRGDJ5CTczRsRCL7)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a0 a0) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ ain (asing a2) a1) → (¬ ain a3 (asing a1)) → (¬ ain (asing a2) a2) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (asm a4 (asing a1)) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain a1 (asing a3)) → ain a0 (aun a1 (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → ain a2 (aun (asing (asing a2)) a4) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 (asm (apow a2) a2) → aal3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal3 a2)
In Proofgold the corresponding term root is f73ea8... and proposition id is e2b51e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZazKYd9vPmpzHTooKE72weSBz6m2ycTYh)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a0 a3 → ain a1 a3 → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (¬ ain a3 a2) → (¬ asubq a3 a2) → (¬ (a0 = a2)) → (¬ (a1 = a2)) → ain a2 (asing a2) → (¬ (a0 = asing (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ (asing a2 = asm (apow a2) a2)) → ain (asing a2) (asing (asing a2)) → (¬ ain a3 (asing (asing a2))) → ain (asing a2) (apow (asing a2)) → ain a0 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a3 (asm a4 (asing a1)) → ain a3 (asm a4 (apow (asing a2))) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (asm a2 (asing a0) = asing a1) → asubq a2 (asm a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → (asm a4 (asing a3) = a3) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal5 (aun (apow a2) (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (apow a2) a2) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))))
In Proofgold the corresponding term root is b87f06... and proposition id is 69f75f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUgMWnzsDeTa4H7p68edo3QV4ABJDSK5Ut)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain (asing a1) a2) → (¬ asubq (asing a1) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow (asing a1)) a3) → (¬ (a0 = apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (asing a2 = a3)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → (asm a2 (asing a1) = a1) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (asm a4 (asing a3) = a3) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal3 (aun a2 (asing (apow a2))) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (asm a4 a2) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))
In Proofgold the corresponding term root is 6cca57... and proposition id is 43cfe1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRwRtnrLcc4QsLeDd6qzmh6CiYhwFR7mHb)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ asubq a2 a1) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ ain (asing a2) a2) → (¬ ain (apow a2) a2) → (¬ ain (apow a2) a3) → (¬ (asing a2 = apow a2)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain a2 (asm a4 (asing a1)) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a2 (aun a1 (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a0 (aun a2 (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a3 (asing a2) = a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (asm (apow a2) (asing (asing a1)) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ asubq a3 a2)
In Proofgold the corresponding term root is c5a0fe... and proposition id is 3ea4b2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHKxNtwkEXsXESp4totVgEcJfb1pD5kSF6)
(∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ (a0 = a3)) → (¬ (asing a1 = a2)) → ain a0 (apow (asing a2)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ (a1 = asing (asing a1))) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (a1 = asm (apow a2) a2)) → (¬ (a1 = apow a2)) → (¬ (a3 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a2) (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aex3 (asm a4 (asing a3)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun a4 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))))
In Proofgold the corresponding term root is 3a90e0... and proposition id is 01bae0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZuSFxCQZvtH6QYKVhbBrusxtMg25p3JQZ)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (¬ ain a2 a2) → (¬ (a2 = a3)) → (¬ ain (asing a2) a1) → (¬ asubq a2 (asing a2)) → (¬ ain (asing a2) a3) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → ain a1 (apow (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing a2) (aun (asing (asing a2)) a4) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal3 (asm a4 (asing a2)) → aal5 (aun (asing (asing a2)) a4) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (apow (asing a2)) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))))
In Proofgold the corresponding term root is ef80a8... and proposition id is 9ffff0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMS5PEKxtRQuK7CFWWtiTvbVkhzrQcN9gSE)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ (a0 = asing a1)) → (¬ ain (asing a1) a2) → (¬ asubq a1 (asing a2)) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (a0 = apow a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a0 (aun a1 (asing a3)) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a2)) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (asm a4 (asing a3) = a3) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 a4) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 (apow (asing (asing a1))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (apow (asing a2)) → aex2 (aun a1 (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))
In Proofgold the corresponding term root is a41450... and proposition id is 165b08...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLAKYFJtTz1At1gQAmkpeTZJgFvZXHPYkQ)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a0 a0) → (¬ asubq a1 a0) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → ain a0 (apow a2) → (¬ ain a0 (asing a1)) → (¬ (asing a1 = a2)) → (¬ asubq a2 (asing a2)) → (¬ (a1 = asing a2)) → (¬ asubq (asing a2) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (a1 = asm (apow a2) a2)) → (¬ (apow (asing a1) = a3)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ (asing a2 = apow a2)) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a2 (aun a1 (asing a3))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aex2 (asm a4 a2) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2))
In Proofgold the corresponding term root is 1d6497... and proposition id is b23626...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVRSZxRPNm3vGqorHQaGH2RMRweyAFwJfV)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a2 a3 → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → asubq (asing a0) a1 → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a1 a1) → ain a0 (apow a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a2 (asm a3 (asing a1)) → ain a0 (apow (asing a2)) → (¬ ain a1 (asing (asing a1))) → (¬ (a2 = asm (apow a2) a2)) → (¬ asubq a3 (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a0 (asm a4 (asing a1)) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a3 (aun a1 (asing a3)) → (¬ ain (apow (asing a1)) a4) → adisjoint (apow (asing a1)) (asm a4 a2) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a2)) a4) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq a1 (asm a2 (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → aal3 (asm (apow a2) (asing a2)) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (asm a3 (asing a1)) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is 35722d... and proposition id is 90df40...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTEEmx9m6Ao8ZAHdtHjAh91wkFmGyi6CC7)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asing a1)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ ain a2 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a3 = asm (apow a2) a2)) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain a2 (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → ain a1 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a1 (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → aal4 (apow a2) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aun (asing a2) (asing (asing a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (apow a2) → aex3 (asm a4 (asing a3)) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))))
In Proofgold the corresponding term root is f3dfa8... and proposition id is 347c4b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUvRRUeFDetaFatYP2WrC9bmtmz86UF3vK)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → asubq (asing a1) a2 → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ ain a1 (asing a2)) → (¬ (a0 = asm a3 (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a3 (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ (a1 = asing (asing a1))) → (¬ ain (asing a2) a2) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ ain (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a3) → (¬ (a0 = apow a2)) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a3 (asm a4 (asing a2)) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal2 (asm a3 (asing a1)) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm a4 a2) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 a4 → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25))))
In Proofgold the corresponding term root is ac9a58... and proposition id is f74e01...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJ6SKTHUPwUz4HdcHKvjmniEQk13zTvapx)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → asubq a3 a4 → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (asing a1) a0) → (¬ ain a0 (asing a2)) → (¬ ain (asing a2) a1) → ain (asing a1) (apow (asing a1)) → (¬ (asing a1 = a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asing a2)) → (¬ (a2 = asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ (a2 = asing a2)) → (¬ ain (asing a2) a3) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a2 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq a3 (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (asm (apow a2) (asing a2)) → aal3 a3 → aal4 (apow a2) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (apow (asing a2)) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 a3 → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (asm a4 (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ asubq (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is 3d56f5... and proposition id is 22d457...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPj4WXDq3uVmLrgR8mKupCxMjDd81bRcxg)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (asing a1) → ain a2 (asing a2) → (¬ asubq a1 (asing a1)) → (¬ (a1 = asing a1)) → asubq a1 (apow (asing a1)) → (¬ ain (asing a1) (asing a2)) → (¬ ain a2 (asing a1)) → ain a2 (apow a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain a3 (asing a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a2 (asm a4 (asing a1)) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (asing a2)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a2 (asm a4 (apow (asing a2))) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a1 (aun (asing (asing a2)) a4) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a3 (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq a1 (asm a2 (asing a1)) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aex2 a2 → aex2 (apow (asing a2)) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun a1 (asing a3)) → aex2 (asm a3 (asing a2)) → aex3 (aun a2 (asing (apow a2))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26)))
In Proofgold the corresponding term root is 45f700... and proposition id is 6c420a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYsTtoSpjcuDF7VXAYHMZZR1Phkf2ZL5Cj)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a3 a2) → (¬ (asing a1 = a2)) → (¬ asubq a2 (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain a3 (apow a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asing a2) (asing a3)) → ain a2 (asm a4 a2) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain a1 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (asm a3 (asing a2)) a2 → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (asm a4 (asing a3) = a3) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun a3 (asing (apow a2))) → aex2 (asm a3 (asing a1)) → aex2 (asm (apow a2) (apow (asing a1))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24))))
In Proofgold the corresponding term root is d1e19a... and proposition id is 305e8d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGMp28iiFKfS8zeamiF2pCVbrvgmHuQdCc)
(∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a2 a2) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ asubq a3 (asing a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (asing (asing a1) = a3)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain a0 (apow (asing (asing a1))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a2 (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (¬ aal2 (asing a2)) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (asm (apow a2) a2) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ asubq (aun (asing (asing a2)) a4) a4)
In Proofgold the corresponding term root is 5d7fb5... and proposition id is ac2182...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSs9482s5BKYGPHtpkoRpqjP23zjy8AHsF)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (¬ asubq a2 (asing a2)) → ain a2 (asm (apow a2) (apow (asing a2))) → ain a0 (apow (asing a2)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (apow a2) a2) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ (a2 = apow a2)) → asubq (asm a3 (asing a1)) (apow a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ (a3 = apow a2)) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain (apow (asing a1)) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a1 (aun a2 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal2 (asm (apow a2) a2) → aal3 (asm (apow a2) (asing a1)) → aal4 (aun a3 (asing (asing a2))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ aal2 (asing a2))
In Proofgold the corresponding term root is 716bc4... and proposition id is 2c0ca8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKrZzDqukFAuU38QUgWsA21RakgUrAWwcV)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, asubq X24 X24) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a0 a0) → ain a0 (apow (asing a1)) → ain (asing a1) (apow a2) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = a3)) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a1 (asm a4 (asing a2)) → ain a3 (asm a4 (asing a2)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a2)) a2 → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (¬ aal4 a3) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal2 (asm a3 (asing a1)) → aal3 (aun (asing a1) (apow (asing a2))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun a3 (asing (apow a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))
In Proofgold the corresponding term root is ecc3a6... and proposition id is 190736...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHSkVn6ztte1kLBD2qHm6igiixqD8QqHfK)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a0) → (¬ ain a3 a2) → asubq a1 a2 → (¬ ain a0 (asing (asing a1))) → asubq a1 (asm a3 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ (a1 = asing (asing a1))) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain a1 (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a2 (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain a0 (asm a4 a2)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) a4 → (¬ aal2 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (asm (apow a2) a2) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm a4 (asing a3)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (∀X24 : set, asubq a0 X24)
In Proofgold the corresponding term root is e35071... and proposition id is 2e9740...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFewfEF1dVG6k4EL86EYLZnsC61rWASTjq)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ ain a2 a1) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (¬ ain (asing a1) a2) → (¬ (a0 = asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain (asing a2) (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a0 (aun a1 (asing a3)) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain a0 (asm a4 a2)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a4 (asing a3)) a3 → (asm a4 (asing a3) = a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal4 a4 → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex4 a4 → aex4 (aun a3 (asing (apow a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq a2 (aun (asing a1) (apow (asing a2)))
In Proofgold the corresponding term root is 2a993c... and proposition id is d90e19...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMM9JdJeDFpoUbsVQu5w9yxYWfhC8w86WDE)
ain a1 a3 → ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (¬ (a0 = a1)) → (¬ ain (asing a1) a0) → ain a1 (apow a2) → ain (asing a1) (asm (apow a2) a2) → (¬ ain a0 (asing (asing a1))) → ain a2 (apow a2) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ ain (apow (asing a1)) a3) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a2 (asm a4 (asing a1)) → ain a2 (aun a3 (asing (asing a2))) → ain a3 (asing a3) → ain (asing a1) (aun (asing (asing a1)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 (asing a3)) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun (apow a2) (asing (apow a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (asm a4 (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (asm a4 (asing a0)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2)))
In Proofgold the corresponding term root is af8e7c... and proposition id is b93271...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZeyzBbiihQa5pCRJNs9SgcFtNSEP3Ai61)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a0 a2 → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ asubq a2 a1) → (¬ ain (asing a1) a0) → ain a0 (apow (asing a1)) → ain a2 (asing a2) → ain a1 (apow a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (a2 = asing (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain a3 (asing a2)) → (¬ (a2 = asing a2)) → (¬ (apow (asing a1) = a3)) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a0 (asm a4 (asing a1)) → (¬ ain (asing a2) a4) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a0 (asm a4 a2)) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 a4) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 (asm (apow a2) a2) → aal3 a3 → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 a3 → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26)
In Proofgold the corresponding term root is 1226f8... and proposition id is 2b1e89...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSmnjVJmZckneVkZemgWfdNmS15D6i5k3u)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a3 a3) → (¬ asubq a4 a3) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ ain a0 (asing a1)) → (¬ asubq a1 (asing a1)) → (¬ ain (asing a1) a1) → (¬ ain a2 (apow (asing a2))) → (¬ ain a1 (asm a3 (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ (a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a2)) → ain a3 (asm a4 (apow (asing a2))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (apow a2)) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))
In Proofgold the corresponding term root is ed1768... and proposition id is 4873b6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJFXbb93ep9FZjTvdkX5hVLvtWm4v9bjDA)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ (a2 = a3)) → ain a0 (apow (asing a1)) → (¬ ain a1 (asing a2)) → ain a2 (asm (apow a2) a2) → (¬ asubq (asing a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a3 (aun a1 (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal4 a4 → aex3 (asm a4 (asing a1)) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex5 (aun (asing (asing (asing a1))) a4) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))
In Proofgold the corresponding term root is 48edda... and proposition id is 58307b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMK1fWA3jv5VtBs3f1fnr5HChHAhQaMn27S)
ain a0 a2 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (¬ asubq a2 a1) → (¬ (a0 = a3)) → (¬ ain a0 (asing a1)) → (¬ asubq a1 (asing a1)) → (¬ (a1 = asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain a2 (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain (apow a2) (apow a2)) → (¬ ain (asing (asing a1)) a3) → (¬ ain (asing a2) (asm (apow a2) a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow a2) a4) → (¬ ain a0 (asm a4 a2)) → ain a2 (asm a4 (apow (asing a2))) → ain a0 (aun (asing (asing a2)) a4) → ain (apow a2) (asing (apow a2)) → ain a1 (aun a3 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asm a3 (asing a1)) a4 → asubq (asm (apow a2) (apow (asing a1))) a4 → (¬ aal2 (asing a2)) → (¬ aal4 a3) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun a3 (asing (asing a2))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ ain (apow (asing a1)) a3)
In Proofgold the corresponding term root is adf53b... and proposition id is f7242b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMV9BG9bMRBntPd1fjzmn73X9Rec3JwrAAt)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → ain (asing a1) (asing (asing a1)) → (¬ asubq a1 (asing a2)) → (¬ ain a2 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a1 = asing (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain a3 (asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (a3 = asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → ain a1 (aun a3 (asing (asing a2))) → ain a0 (asm a4 (asing a2)) → (¬ ain (apow a2) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a3 (asm a4 (asing a1)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm a3 (asing a2)) a2 → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal4 a3) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 a2 → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (apow (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (apow a2) → aex5 (aun (apow a2) (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1))
In Proofgold the corresponding term root is 3a1fd8... and proposition id is fc39b8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMS4Giszsrig24tQTFS766HZmhKch7c8GcE)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → asubq a2 a3 → ain a2 (asing a2) → (¬ ain a1 (asing a2)) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq a3 (apow a2) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a0 (asm a4 (asing a2)) → (¬ ain (asm a3 (asing a1)) a4) → ain a2 (asm a4 a2) → ain a3 (asm a4 (asing a1)) → ain a3 (asm a4 (apow (asing a2))) → ain a1 (aun (asing (asing a2)) a4) → ain a0 (aun a1 (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aal3 a3 → aal3 (asm (apow a2) (apow (asing a2))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (apow a2) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex3 (asm (apow a2) (asing (asing a1))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2))
In Proofgold the corresponding term root is 1ff353... and proposition id is aa0d78...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPWiUk5vak8WFrLp8mTZGPhn4NEas5uKTH)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (¬ ain a0 a0) → (¬ ain (asing a1) (asing a2)) → (¬ ain a3 (asing a1)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing a2)) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun a2 (asing (apow a2))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun (apow a2) (asing (apow a2))) → aex3 (asm a4 (asing a2)) → aex3 (asm a4 (asing a0)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25)))
In Proofgold the corresponding term root is 28296e... and proposition id is 796e7b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMX81LTRSVYTiQ3ksAFi7bSVxpiqn5xCRw8)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a3 a2) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ ain (asing a1) a2) → (¬ ain a1 (asing a2)) → (¬ (a0 = asing a2)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ asubq a2 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain a2 (apow (asing a2))) → (¬ ain (apow a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (asing a1 = apow a2)) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (a2 = apow a2)) → (¬ ain (asing a2) a3) → (¬ (asm a3 (asing a1) = apow a2)) → ain a1 (apow (apow (asing a1))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a0 (asing a3)) → (¬ ain a2 (asing a3)) → ain a1 (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a1) = a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aex2 (asm (apow a2) a2) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ (a1 = asing (asing a1)))
In Proofgold the corresponding term root is 9ae066... and proposition id is e77f66...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMR5adY1ZSb9dx8Asn3iu2XkFHhRqrFVS8L)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (¬ ain X24 a0)) → ain a1 a2 → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a2 a0) → (¬ asubq a2 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a0 (apow a2) → (¬ asubq a1 (asing a2)) → (¬ asubq a2 (asing a1)) → ain a0 (apow (asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain a3 (apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a3 (aun (asing a1) (asing a3)) → ain a3 (asm a4 (asing a1)) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → ain (apow a2) (aun a4 (asing (apow a2))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a4 (asing a3)) a3 → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (asm a4 a2) → aal3 (aun (asing a1) (apow (asing a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (apow a2) (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm a3 (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a1)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3)))
In Proofgold the corresponding term root is df0f40... and proposition id is 415f00...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZR5SSJKx4cJjMAuuPrKscVAFzynJE2DT6)
ain a2 a3 → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ asubq a3 a2) → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ asubq a1 (asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asing a2)) → (¬ (a0 = apow a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain (asing a2) (asing a3)) → (¬ ain (asing (asing a1)) a4) → (¬ ain (asing a1) (asm a4 a2)) → ain a1 (aun (asing (asing a2)) a4) → ain a1 (aun a3 (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asing a1) (asm a2 (asing a0)) → asubq a1 (asm a2 (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a4 (asing a3)) a3 → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal2 (asm a3 (asing a1)) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ (a1 = a3))
In Proofgold the corresponding term root is f2e81d... and proposition id is fbc13e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcmBPSBaokAZU6RRmLaQ4rCGEhGty2MZ4n)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → asubq a1 a2 → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a2 (asing a2) → (¬ ain a1 (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a2) a4) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain (asing (asing a1)) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a1 (asm a4 (apow (asing a2))) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a1 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → aal3 a3 → aal5 (aun (asing (asing (asing a1))) a4) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25)))
In Proofgold the corresponding term root is 54b2bb... and proposition id is 600184...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKdWs6ShgLZSa8tFm986UKiL2yymVSmyTe)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a0 a2 → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a1) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a0 (asm a3 (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ asubq a2 (asing a2)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a1 = apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → ain a2 (asm a4 (asing a1)) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a0 (asing a3)) → ain a3 (aun (asing a1) (asing a3)) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a2)) a4) → ain (apow a2) (asing (apow a2)) → ain a1 (aun a2 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun (asing (asing a1)) a4) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 (asing a3)) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → aal3 (asm (apow a2) (asing a1)) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (aun (asing (asing a2)) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25))
In Proofgold the corresponding term root is 34b1fa... and proposition id is 9aa31d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFwAFuxBAaGFP9q7HXs8knwNQyawGTM8Jx)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ asubq a2 a1) → (¬ (a0 = a1)) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ ain (asing a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain a3 (asing a2)) → ain a0 (apow (asing (asing a1))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a1 (apow (asm a3 (asing a1))) → ain a0 (aun a1 (asing a3)) → ain a2 (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a2) (aun (asing (asing a2)) a4) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a3 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → ain a2 (aun a3 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) a4 → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (apow a2)) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aex2 (aun a1 (asing a3)) → aex2 (aun (asing (asing a1)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq a3 (aun a3 (asing (asm (apow a2) a2)))
In Proofgold the corresponding term root is 35054f... and proposition id is 5b8aeb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMStnzotb4uwJZZ7pQRvwTcBGx8sDUeLEou)
(∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a2) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a0 (apow a2) → (¬ asubq a1 (asing a1)) → (¬ ain a1 (apow (asing a1))) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asm (apow a2) a2)) → ain a0 (apow (asing a2)) → (¬ ain (asing a2) a2) → (¬ ain a3 (asing a2)) → (¬ ain (asing a2) a3) → asubq a3 (apow a2) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (apow a2) (asing (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a3 (asing a3) → (¬ ain a2 (aun a1 (asing a3))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq a3 (asm a4 (asing a3)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → aal3 a3 → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm a3 (asing a1)) → aex2 (aun (asing (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 a4 → aex5 (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)))
In Proofgold the corresponding term root is 2122e0... and proposition id is f3f372...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJ1jSRzEXR4QedURWpJbztVhoVztj8RjUd)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a2) → (¬ ain a3 a2) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (asing a1 = asing a2)) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (asing (asing a1)) a3) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (asm a3 (asing a1) = a3)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing (asing a1)) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → (¬ ain a0 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a4 (asing a3)) a3 → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing a1)) a4) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → aal2 (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is 50f905... and proposition id is 0b7309...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXKRX5xf9FFomGQyxYSbZuUZeLYN7Gm7Y5)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ (a1 = a2)) → (¬ (a1 = asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ (a0 = apow (asing a1))) → (¬ (asing a1 = a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ (asing a1 = a3)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (asing a2) a3) → asubq (asm a3 (asing a1)) (apow a2) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (asing a2)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a0 (aun (asing (asing a2)) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a4 (asing a3)) a3 → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → aal4 (apow a2) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (asm a3 (asing a2)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm a3 (asing a1)) a4
In Proofgold the corresponding term root is a82dac... and proposition id is a746cf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZsfjWBHFhcLToAcmN4WhdXGkxAZhVxxzk)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (¬ ain a1 a1) → (¬ (a0 = a1)) → ain a1 (asing a1) → ain (asing a1) (apow a2) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → ain a2 (asm a3 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = a3)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (a2 = asing a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asing a2) (asing a3)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal2 (asm a3 (asing a1)) → aal5 (aun a4 (asing (apow a2))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asing a1) (asm (apow a2) (asing a2))
In Proofgold the corresponding term root is 494707... and proposition id is 99224a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFcfB83YtRXsqtbZDeDqyZzkZUt4LVAjHU)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → asubq (asing a0) a1 → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a2) a1) → (¬ ain (apow a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a2 (asm a4 a2) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → aal2 (asm (apow a2) a2) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aun a1 (asing a3)) → aex3 (asm (apow a2) (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (asm (asm a4 (asing a2)) (asing a3) = a2)
In Proofgold the corresponding term root is f70756... and proposition id is 443efa...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWrWabbpSKyUZTLBNu2H33bHTfnLfQvSKy)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → asubq a3 a4 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ ain (asing a1) a0) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (apow (asing a2))) → asubq a1 (apow (asing a1)) → ain a2 (asm a3 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ (a2 = apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) a3) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asing (asing a1) = a3)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain (asing a2) (apow (asing a2)) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow (asing a1)) a4) → (¬ ain (asm a3 (asing a1)) a4) → ain a3 (asm a4 (asing a1)) → (¬ ain a1 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow a2) (asing a1)) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (aun a3 (asing (apow a2))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (apow (asing a2)) → aex2 (aun a1 (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2)
In Proofgold the corresponding term root is a0fdbb... and proposition id is 92a9ec...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPQcsGRPnzr3o7YQos8nRZVS1hYuDzLaE7)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a1 a1) → asubq a2 a3 → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asing a2)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → ain a2 (asm (apow a2) a2) → ain a0 (apow (asing a2)) → (¬ asubq (asing a1) (asing (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ (a2 = apow (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a2 = apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain a0 (asing (apow (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain (asing a1) (aun (asing (asing a1)) a4) → (¬ ain a0 (asm a4 a2)) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a2 (asing a1) = a1) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → aal3 (asm (apow a2) (asing a1)) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)))
In Proofgold the corresponding term root is 4bd7d7... and proposition id is 8c56d2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHvmSVqdzgt5QHAqB5TnJC3iPDHZZvbExC)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ asubq a3 a2) → (¬ (a0 = asing a1)) → (¬ (a0 = asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a2 (asing a3)) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing a2)) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) a4 → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a2)) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal2 (apow (asing (asing a1))) → aal3 (aun a2 (asing (apow a2))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun a1 (asing a3)) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ (asing a1 = a2))
In Proofgold the corresponding term root is 6a2fec... and proposition id is d892ae...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZ5so3Qdk2MqUH29RYgpsni2QyXUedFqrN)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain a0 (asing a2)) → (¬ (a0 = apow (asing a1))) → asubq a1 (apow (asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = a2)) → (¬ ain a2 (apow (asing a2))) → (¬ (asing a1 = asm a3 (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain a3 (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a3) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain (asing a2) (apow (asing a2)) → (¬ ain a3 (apow (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a0 (asm a4 (asing a2)) → (¬ ain (asm a3 (asing a1)) a4) → ain a3 (asm a4 (apow (asing a2))) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → aal3 a3 → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2))
In Proofgold the corresponding term root is 6432d9... and proposition id is 8e6791...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcTsYXi1rjDb947YJhqKdPpPegYX2GU6An)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → (¬ asubq a2 a1) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a0 (apow a2) → (¬ (a0 = asing (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (apow a2) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → asubq (asm a3 (asing a1)) (apow a2) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a2) a4) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain (apow a2) a4) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm a2 (asing a0)) (asing a1) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 a4) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm (apow a2) a2) → aal2 (asm a4 a2) → aal4 (aun a3 (asing (apow (asing a1)))) → aex2 (asm a3 (asing a1)) → aex3 (asm a4 (asing a1)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1))
In Proofgold the corresponding term root is feddc3... and proposition id is db65b4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHDQRhqKA87vHRm8TnFNAHwy1jLMf9LHuG)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a0 (asing a2)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ asubq a2 (asing a1)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ asubq a3 (asing a2)) → (¬ (asing (asing a1) = a3)) → (¬ (asing a2 = a3)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → (¬ ain a0 (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a4 (aun (asing (asing a1)) a4) → (asm a2 (asing a0) = asing a1) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) (asing (asing a1))) a3 → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → (¬ aal3 (apow (asing a2))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 a2 → aal3 (asm (apow a2) (asing a2)) → aal3 (asm (apow a2) (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → aex4 (apow a2)
In Proofgold the corresponding term root is edd713... and proposition id is 9d76a5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQoYyubsWDtumy9eiGsYiPCqhBG6mVmknE)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ asubq a1 (asing a1)) → (¬ (a0 = asing a1)) → (¬ (a1 = asing a1)) → (¬ ain a0 (asing (asing a1))) → asubq a1 (asm a3 (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ (a1 = asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asing a2) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a0 (asing a3)) → (¬ ain a1 (asing a3)) → ain a0 (aun a1 (asing a3)) → ain a2 (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain a2 (aun a3 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (asm (apow a2) (asing (asing a1))) a3 → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing (asing a1))) → (¬ aal2 (asing a3)) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal3 (asm a4 (asing a1)) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (apow a2) (asing (apow a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (asm a4 a2) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is 7ae7a6... and proposition id is e80a57...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXZiXJ9f4B2HD4ZMjMrdFRHQjh41RhhjVV)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a2 a0) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ (a0 = asm a3 (asing a1))) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) (asing a2)) → ain a2 (asm a3 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asing a2)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (asing (asing a1) = a3)) → (¬ (asing a2 = a3)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain a2 (asm a4 (asing a1)) → ain a0 (aun a3 (asing (asing a2))) → ain a3 (asing a3) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1))))
In Proofgold the corresponding term root is 2b0623... and proposition id is 30698d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPu7v7LAbKUSmnfB9QhPbUYUuFszGr5E9H)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (¬ asubq a2 a0) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ (a0 = asing a1)) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) a3) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a0 (apow (apow (asing a1))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain a0 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (¬ aal3 (apow (asing a1))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun a3 (asing (asing a2))) → aex3 (asm a4 (asing a0)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ (a0 = apow a2))
In Proofgold the corresponding term root is ac92f2... and proposition id is 0faaf6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMP7La6D93yF58tnwCSM3KDDdX2Mx6VmQm9)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a0 a1 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → asubq a3 a4 → ain a2 (asing a2) → (¬ ain (asing a1) a2) → (¬ (a0 = apow (asing a1))) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ asubq a2 (asing a1)) → (¬ (a1 = asing (asing a1))) → (¬ ain (asing a2) a2) → (¬ asubq a2 (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a0 (asm a4 a2)) → ain a2 (asm a4 a2) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal5 a4) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 a2 → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm a4 a2) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25)))))
In Proofgold the corresponding term root is 542394... and proposition id is b5c5d6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNy3CT4VEGkmRzy2yJzrW3CWAp3nbCTLcH)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a2 a4 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (¬ (a0 = asing a1)) → ain (asing a1) (asm (apow a2) a2) → (¬ ain a0 (asing (asing a1))) → (¬ asubq a2 (asing a1)) → (¬ ain a2 (apow (asing a1))) → (¬ (a2 = apow (asing a1))) → (¬ asubq a3 (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a1 (asm a4 (apow (asing a2))) → ain a1 (aun (asing (asing a2)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 a1) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → aal2 (asm (apow a2) a2) → aal3 (asm a4 (asing a2)) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq a1 (asm a3 (asing a1))
In Proofgold the corresponding term root is 58ca64... and proposition id is 1924ee...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYaVXKmCgubXRvjSB7xoMynLyz6eA8CFvp)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a0 a2 → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a2 a2) → ain a1 (asing a1) → ain (asing a1) (asing (asing a1)) → ain a2 (asing a2) → (¬ ain (asing a1) a1) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a2 (asm a3 (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ (a1 = asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asing a1) a3) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → ain a2 (asm a4 a2) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a2)) a2 → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 a1) → (¬ aal2 (asing (asing a1))) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (apow (asing (asing a1))) → aal2 (asm a4 a2) → aal5 (aun (asing (asing a2)) a4) → aex2 a2 → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24)
In Proofgold the corresponding term root is 8279b5... and proposition id is 3009ba...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLuAV9ry1sV6qvLDzGGQSD9vifYxCfWRUz)
ain a2 a3 → ain a0 a4 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ asubq a1 a0) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain a1 (apow (asing a2))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ ain a3 (apow a2)) → (¬ asubq a3 (asing a2)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (asing a3)) → (¬ ain (apow a2) a4) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain a3 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm a3 (asing a2)) a2 → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 a3) → (¬ aal5 (aun a3 (asing (apow a2)))) → aal4 a4 → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun a1 (asing a3)) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2))))
In Proofgold the corresponding term root is 76f045... and proposition id is 3deb9c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHZgaMRkD49uxgLAzLByhz9rtda9rBGmo4)
(∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → asubq (asing a1) a2 → (¬ asubq a1 (asing a2)) → (¬ ain a1 (asing a2)) → (¬ asubq a2 (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ (a1 = asing a2)) → (¬ (a1 = apow (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (aun (asing a1) (apow (asing a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (asm a3 (asing a1)) a2)
In Proofgold the corresponding term root is 921d9e... and proposition id is 5114cc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPmpbiGqooasuVUsWphdWCbnAruH3pNzmW)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (¬ asubq a3 a2) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain (asing a1) a0) → ain (asing a1) (asing (asing a1)) → ain a0 (apow (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = a3)) → (¬ (asm (apow a2) a2 = apow a2)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain a0 (asing a3)) → (¬ ain (asing (asing a1)) a4) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a2 (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq a1 (asm a2 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (aun (asing a1) (apow (asing a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2)))
In Proofgold the corresponding term root is ea9c05... and proposition id is edcf3d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJ87usMyRhXDjmJWAb5rC6xwWm4x67QiK6)
(∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a1 a0) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ (asing a1 = a2)) → (¬ asubq a2 (asing a1)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain a3 (apow a2)) → (¬ ain (apow a2) (asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a2 (asing a3)) → ain a0 (asm a4 (asing a2)) → ain a3 (asm a4 (apow (asing a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (asm a4 (asing a2)) → asubq a4 (aun a4 (asing (apow a2))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a1 (asm a2 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 a3) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 a2 → aal2 (asm a4 a2) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal4 (aun a3 (asing (apow a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (asm a4 a2) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (aun (asing a2) (apow (asing a2))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → ain a1 (apow (asm a3 (asing a1)))
In Proofgold the corresponding term root is a9cbc1... and proposition id is 3b1802...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQ9vYEkcCPGpaFHjLfbDf1x4agnCJPYeup)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (¬ ain a2 a1) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a2 (asing a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a2 (asm a3 (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) a3) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (apow (asing a1) = a3)) → ain a0 (apow (asing (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a1 (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun a2 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (apow (asing (asing a1))) → aal4 (apow a2) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (apow a2) a2) → aex3 (asm a4 (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24))))
In Proofgold the corresponding term root is 48bad6... and proposition id is b0dc15...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMP4KJsmyNm5kBniFPbgbsPAo8W1muZxPT7)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a0 a4 → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a0 a0) → ain (asing a1) (asing (asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a2 (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ (a1 = asing (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → asubq a3 (apow a2) → ain a0 (apow (asing (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain (apow a2) a4) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a1 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 a3) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 a2 → aal2 (asm (apow a2) a2) → aal4 (apow a2) → aex4 (apow a2) → aex4 (aun a3 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is f2cff5... and proposition id is a4b3c7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMT1jQuHycrWf4E1h65AaUbDnTksh8NpuyX)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a2 a2) → asubq a1 a2 → (¬ asubq a4 a3) → (¬ (a0 = a1)) → (¬ (a2 = a3)) → ain a1 (apow a2) → asubq (asing a1) a2 → (¬ ain a0 (asing (asing a1))) → (¬ asubq a2 (asing a2)) → (¬ ain a3 (asing a1)) → (¬ (a1 = apow (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a0 (asm a4 (asing a1)) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a3 (aun a1 (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → (¬ ain a0 (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (asm (apow a2) a2) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asing a2) (asm (asm a4 a2) (asing a3))
In Proofgold the corresponding term root is 1c2f80... and proposition id is c15876...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMU4TZeCaR8KnqjJv3yTt3Vosp2VRj63hs6)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (¬ asubq a2 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ ain a1 (apow (asing a1))) → ain (asing a1) (apow a2) → (¬ asubq a1 (asing a2)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) a3) → asubq a3 (apow a2) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (apow (asing a2))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asing (asing a1)) a4) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25)))
In Proofgold the corresponding term root is b41709... and proposition id is 02a747...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHy5FJMwb3yM2anhFX7W4oN9AmUSCsLSaD)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ asubq a1 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a1 (asing a2)) → ain a2 (asm (apow a2) a2) → ain a0 (apow (asing a2)) → (¬ (a1 = apow (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ (a3 = apow a2)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (asing a3)) → ain a0 (asm a4 (asing a2)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a3 (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 a2) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal2 (asm a4 a2) → aal3 (aun a2 (asing (apow a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (asm (asm a4 a2) (asing a2) = asing a3)
In Proofgold the corresponding term root is ab1370... and proposition id is 19d899...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKGyFFp6Hz3oiZTe9SnYAvr2bVrvrhmSis)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ ain a3 a3) → ain a1 (asing a1) → (¬ ain a0 (asing a1)) → ain (asing a1) (apow a2) → (¬ (a0 = asing a2)) → (¬ ain (asing a2) a1) → ain a2 (asm (apow a2) a2) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (asing a1 = asing (asing a1))) → (¬ (a2 = apow a2)) → (¬ (asm a3 (asing a1) = a3)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain (asing a1) (aun (asing (asing a1)) a4) → (¬ ain (asing a1) (asm a4 a2)) → ain a0 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing (asing a1))) → (¬ aal3 a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal2 (asm a4 a2) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun a3 (asing (apow a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm a3 (asing a2)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) a4)
In Proofgold the corresponding term root is 2b4024... and proposition id is c8b49a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMb8fJjBDSehsLTLZrrRKHJMmtDEqrsfxtN)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (¬ ain a1 a1) → (¬ ain a0 (asing a2)) → (¬ asubq a1 (asing a1)) → (¬ ain a1 (apow (asing a1))) → (¬ ain (asing a2) a1) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain (apow a2) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (apow (asing a1) = a3)) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (asm a4 (asing a1)) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a1 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a0 (aun a3 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 a1) → (¬ aal2 (asing a2)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm (apow a2) a2) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm (apow a2) a2) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3)))
In Proofgold the corresponding term root is 2cc446... and proposition id is 7b271a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHHtq7gv62a1vrm8fZxPAjUTcfVFfFsGqf)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a2 a0) → asubq a1 a2 → (¬ (a0 = a3)) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a0 (asm (apow a2) (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (apow a2) a2) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (asing a2) a3) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (a3 = apow a2)) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a2 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a1 (asm a4 (apow (asing a2))) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain (asing a2) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq a3 (asm a4 (asing a3)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (asing (asing a2)) a4) → aex2 (asm (apow a2) a2) → aex2 (aun a1 (asing (apow a2))) → aex3 (aun a2 (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25))
In Proofgold the corresponding term root is 087a53... and proposition id is 19429b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYipg3FT4DEfJJJivyYZJH9heC2DaFTHF3)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (¬ ain a3 a3) → (¬ (a0 = a3)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ (a1 = asing a1)) → asubq a1 (asm a3 (asing a1)) → ain a0 (apow (asing a2)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (a1 = asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ asubq a3 (asing a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain a0 (apow (apow (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (aun a1 (asing a3))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a0 (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a2 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 a2 → aal2 (apow (asing (asing a1))) → aal3 a3 → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aun a1 (asing (apow a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (aun (asing (apow (asing a1))) a4) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain a0 (apow X24))
In Proofgold the corresponding term root is 41c034... and proposition id is f2d947...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNx9J6CcCjCiZEMyUGxCeQXrWsZHvMgCvN)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain (apow a2) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (asing (asing a1) = a3)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm a4 (asing a1)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun (apow a2) (asing (asing a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (asm (apow a2) a2) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1))))
In Proofgold the corresponding term root is 5c808b... and proposition id is 0d34ef...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPc4imYdB4LXCDC13Z2HtpHDS7C3Re4AT7)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a2 a1) → asubq a3 a4 → (¬ (a0 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (¬ ain a1 (apow (asing a2))) → ain a1 (asm (apow a2) (asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (a1 = asing (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (apow a2) (apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ asubq (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow (asing a1)) a4) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a3 (aun a4 (asing (apow a2))) → ain a2 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq a1 (asm a2 (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (aun a2 (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (apow a2) (apow (asing a1))) → aex3 (asm a4 (asing a2)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex5 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (apow X24))
In Proofgold the corresponding term root is 28cc32... and proposition id is ea4184...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJWgqRsFRAa9eD3ENzHYQ7UagKpvzCACSM)
(∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ asubq a4 a3) → (¬ asubq a2 a0) → (¬ ain a1 (apow (asing a1))) → (¬ (a0 = apow (asing a1))) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (¬ asubq a3 (asing a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a1 (asm a4 (asing a2)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq a3 (asm a4 (asing a3)) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → aal2 (asm (apow a2) a2) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2))))
In Proofgold the corresponding term root is 16afe9... and proposition id is efbc77...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNBN66Tzk2uf7TxPjBfiDvy5ziAEoXWMFj)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a1 a0) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (asing a1) → (¬ (a0 = asing a1)) → ain (asing a1) (asm (apow a2) a2) → (¬ ain a1 (apow (asing a2))) → ain a2 (apow a2) → (¬ (a1 = asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = apow (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asing a2) a3) → (¬ ain a3 (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (a3 = apow a2)) → (¬ (asm (apow a2) a2 = apow a2)) → (¬ ain a0 (asing (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (asm a4 (asing a1)) → (¬ ain (asing a2) a4) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a3 (asing a3) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain (asing a2) (aun (asing (asing a2)) a4) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a1 (asm a2 (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) a4 → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aex2 (asm a4 a2) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1)))
In Proofgold the corresponding term root is 749eee... and proposition id is cf70ae...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXaKVJdi6RBqLKa38GJNQXaqqMwbpGNPqw)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → ain a0 (apow (asing a1)) → asubq (asing a1) a2 → (¬ (a0 = asing a1)) → ain (asing a1) (asm (apow a2) a2) → (¬ ain (asing a2) a1) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain a1 (asing (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (a2 = apow (asing a1))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal5 (aun (apow a2) (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 a3 → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 a4 → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → aex2 (asm (apow a2) (apow (asing a1)))
In Proofgold the corresponding term root is ac141e... and proposition id is 56d0ec...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVwGAVXSf2bSDDtRGu1ir7yHWmT44xkCRp)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a0 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain a2 (asing a2) → (¬ ain (asing a1) (asing a2)) → (¬ ain a2 (apow (asing a2))) → (¬ ain (asing a2) a2) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain a3 (asing a2)) → (¬ (apow (asing a1) = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal3 (asm (apow a2) (asing a1)) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (asm a3 (asing a1)) → aex2 (apow (asing a2)) → aex3 a3 → aex3 (asm a4 (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → ain a1 (asm (apow a2) (apow (asing a1)))
In Proofgold the corresponding term root is 8924d6... and proposition id is 298c66...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXYvWKwZGwNY4zwGb8LRpojbj6CGatSm3X)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a0 a4 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → asubq a1 (asm a3 (asing a1)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq a2 (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow (asing a1)) a3) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal2 (asm a4 a2) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a1)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1))))
In Proofgold the corresponding term root is 4b141d... and proposition id is a31b8d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdWUcoZwR6ReDxZqQNxhBTtWJhQPd13h5q)
(∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a0 a3 → ain a3 a4 → (∀X24 : set, asubq a0 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (¬ ain a1 (apow (asing a1))) → (¬ (a0 = asing (asing a1))) → (¬ asubq a2 (asing a2)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (apow a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ ain (asing (asing a1)) a3) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (a3 = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a0 (apow (asing (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a2) a4) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (apow a2) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain a0 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (asm a4 a2)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal3 a3 → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (apow (asing a1)) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3)))
In Proofgold the corresponding term root is 303cef... and proposition id is 09b23c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYeoCtScMmL7JS7oF1dAp2DJYSCeDiKENc)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (¬ ain X24 a0)) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (¬ ain a2 a2) → asubq a1 a2 → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ ain a0 (asing a2)) → ain a1 (apow a2) → ain (asing a1) (apow a2) → (¬ (a1 = asing a1)) → ain a2 (asm (apow a2) a2) → (¬ (asing a1 = asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (a2 = apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (asing a2) a4) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun a4 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm a2 (asing a1) = a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → aal2 a2 → aal3 (asm a4 (asing a1)) → aal5 (aun (asing (asing a2)) a4) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aun (asing (asing a1)) (asing a3)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (aun (apow a2) (asing (apow a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2)))
In Proofgold the corresponding term root is dc971d... and proposition id is 4969ad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLp3maGiFgq8wA92GLp32AjiwMikN2Ghrr)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → asubq a2 a3 → (¬ asubq a1 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a1))) → (¬ asubq a2 (asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ ain (apow a2) (asing (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) a3) → (¬ (apow (asing a1) = a3)) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → ain a2 (asm a4 a2) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a0 (aun a2 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (asm a4 (asing a2)) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → (asm a2 (asing a1) = a1) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (¬ aal2 (asing a2)) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow a2) (asing a2)) → aal3 a3 → aex2 a2 → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0))
In Proofgold the corresponding term root is 23f5b1... and proposition id is d36fd8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMb8FWjqYtcGSmdNFPfFMnirUuNUW7H3NbJ)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a1 a4 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ (a0 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (¬ (a0 = asing (asing a1))) → ain a2 (asm (apow a2) (apow (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asing a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ (a0 = apow a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a2 (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a2)) a4) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ ain a2 (asing a1))
In Proofgold the corresponding term root is 9a5a93... and proposition id is 71ee5c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJ2UZX2PETkbvGtW3jksV5KXiznxKD3J98)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → ain a0 a2 → ain a0 a4 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (¬ ain a1 a0) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a1 (apow a2) → (¬ asubq a1 (asing a1)) → asubq (asing a1) a2 → ain a1 (asm (apow a2) (apow (asing a2))) → asubq a1 (apow (asing a1)) → (¬ (a1 = apow (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a2 = asm (apow a2) a2)) → (¬ (a2 = asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (asm a3 (asing a1) = a3)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → ain a1 (apow (apow (asing a1))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (apow (asing a1)) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (asm (apow a2) (asing a2)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1)
In Proofgold the corresponding term root is 3f1a42... and proposition id is bfab96...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSxMxXBdSM3se5vaF6b9gUMDVbACPAbHVA)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a1 a1) → (¬ ain a2 a2) → (¬ (a0 = a2)) → ain a0 (asm (apow a2) (asing a2)) → (¬ asubq a1 (asing a2)) → (¬ ain a1 (apow (asing a2))) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = a3)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (apow (asing a1) = a3)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a2) a4) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a0 (asing a3)) → (¬ ain a1 (asing a3)) → ain a3 (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq a2 (asm a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq a3 (asm a4 (asing a3)) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → aal3 a3 → aex3 (asm a4 (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm a4 (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))
In Proofgold the corresponding term root is 46e171... and proposition id is bff8d0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbf4okrQ4T3dCnHfXEAF8xE5gLJ6XR3M77)
(∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ (a0 = a3)) → (¬ ain a0 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → asubq a1 (apow (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ (a1 = asm a3 (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain a3 (asing a2)) → (¬ (a2 = apow a2)) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing (asing a1)) a4) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a2 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 a3) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 a4) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (asing (asing (asing a1))) a4) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex5 (aun (apow a2) (asing (asing a2))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ ain (asing a1) (asing a2))
In Proofgold the corresponding term root is b557db... and proposition id is 7ca035...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQS15gUiZ8Sn9J4gpi9qVMcpKbiwomh1nz)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a2 (asm (apow a2) a2) → ain a2 (asm (apow a2) (asing a1)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asing a2) a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a0 (apow (asm a3 (asing a1))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a2 (asm a4 a2) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asing a2) (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal2 (asm a3 (asing a1)) → aal2 (asm (apow a2) a2) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (asm a3 (asing a2)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun a3 (asing (asing a2))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex5 (aun a4 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2))))
In Proofgold the corresponding term root is a5e970... and proposition id is 6168d2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNeD5KEtQi2Q1SvhAXKhALdyHbwAZs4AE9)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ asubq a2 a1) → (¬ asubq a4 a3) → (¬ asubq a1 (asing a2)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (apow a2) (asing a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ ain (asing a2) (asm (apow a2) a2)) → ain a0 (apow (asing (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a0 (aun a1 (asing a3)) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (asm a2 (asing a1)) a1 → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 a4) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 a4 → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal5 (aun (asing (asing a2)) a4) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun a3 (asing (asing a2))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2)))
In Proofgold the corresponding term root is 5b404c... and proposition id is 88f562...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTWXrjnwBkY9a4onpGkTECNBBbaZRXxL1W)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ asubq a2 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (asing a2)) → ain a0 (asm a3 (asing a1)) → (¬ ain (asing a1) (asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (apow (asing a1)) a3) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (asing (asing a1) = a3)) → asubq a3 (apow a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asing a2) a4) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a1 (asm a4 (asing a2)) → (¬ ain (asm (apow a2) a2) a4) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 a2 → aal3 (asm (apow a2) (apow (asing a2))) → aal4 a4 → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (apow a2) (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3)))
In Proofgold the corresponding term root is 801a3f... and proposition id is 0511a3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSFYNUVDq3GPd2ywyEEjh5cZTgtxXD4a6S)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ asubq a3 a2) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a2 (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ (a2 = asing a2)) → (¬ (apow (asing a1) = a3)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a1) a4) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (asm a2 (asing a0)) (asing a1) → (asm a2 (asing a0) = asing a1) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq a3 (asm a4 (asing a3)) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (asm (apow a2) (asing (asing a1)) = a3)
In Proofgold the corresponding term root is 4595d7... and proposition id is d5a979...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMF7tfcqFdwotyeHoHmEroa3Eenn1kj8qpG)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (¬ (a1 = a3)) → ain a1 (apow a2) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ asubq a3 (asing a2)) → (¬ (asing a2 = a3)) → (¬ (asing a2 = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a3 (aun (asing a1) (asing a3)) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (asm (apow a2) (apow (asing a1))) → aex3 (asm (apow a2) (asing a2)) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (apow a2) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → ain a0 (aun (apow (asing a2)) (asing (apow a2)))
In Proofgold the corresponding term root is 4ae701... and proposition id is 56f50a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHW18iLeC46ZvdQCxQVGvgFVvrM1ABbXCV)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ asubq a4 a3) → (¬ asubq a1 (asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a1 (apow (asing a2))) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a2 (asing a1)) → ain a2 (asm a3 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ ain (asing a1) (apow (asing a2))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ asubq (asing a2) a2) → (¬ asubq (asm a3 (asing a1)) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a1 = asm (apow a2) a2)) → (¬ (a1 = apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (asing a2 = a3)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → ain a1 (apow (asm a3 (asing a1))) → ain a2 (asm a4 (apow (asing a2))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (asing (apow a2)) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (asm a2 (asing a0) = asing a1) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (asm a4 (asing a1)) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))
In Proofgold the corresponding term root is 60ab4a... and proposition id is 30c99d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcRDr9KgM6utLHKoxhu2ErFmCcSkDpTnTE)
ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ (a0 = a2)) → (¬ (a1 = a2)) → (¬ (a1 = a3)) → asubq a1 (asm a3 (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a2) a4) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → ain a3 (asm a4 (apow (asing a2))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 a3) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is bc0f1b... and proposition id is 39983c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUawx1eoFAtu8p2eXt64dBWYYnojBxgacV)
(∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (¬ ain a3 a3) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → ain (asing a1) (asm (apow a2) (asing a1)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain a3 (apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain a0 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a4 (asing a3)) a3 → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aun (asing a2) (apow (asing a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ (asing a1 = asing (asing a1)))
In Proofgold the corresponding term root is 211d1f... and proposition id is b7b876...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPnhjHJFbMvpeP83yk8kwc8MJm3rWrGFiN)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → asubq a1 a2 → ain a0 (asm (apow a2) (asing a2)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (apow a2) a3) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) a4) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a0 (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a0 (aun (asing (asing a2)) a4) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (¬ aal3 (apow (asing a2))) → (¬ aal5 (apow a2)) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun a4 (asing (apow a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm a4 a2) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 a4 → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ ain (apow a2) (apow a2))
In Proofgold the corresponding term root is 1d21cc... and proposition id is 514cef...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdr8MSVbB7QiiVb9DsS7it23SBSJPsomzX)
(∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → asubq a1 a2 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a0 (asm a3 (asing a1)) → (¬ asubq a1 (asing a1)) → (¬ ain (asing a2) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq (asing a2) a2) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ (a1 = apow a2)) → (¬ (asing (asing a1) = a3)) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal4 (apow a2) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 a2 → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (asm a4 (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3))
In Proofgold the corresponding term root is dcd4df... and proposition id is b23024...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVBbFgnc3pNyrc9VMKgpvQVAFnnTTKTu5h)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ asubq a2 a0) → ain a1 (asing a1) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ asubq a1 (asing a2)) → (¬ asubq a2 (asing a2)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asing a2)) → (¬ ain (asing a2) a2) → (¬ asubq (asm (apow a2) a2) a2) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ ain (apow (asing a1)) a3) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a1 (aun (asing a1) (asing a3)) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → asubq a3 (aun a3 (asing (asing a2))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm a4 (asing a3) = a3) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing (asing a1))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun a3 (asing (apow a2))) → aex3 (asm (apow a2) (asing a1)) → aex3 (asm a4 (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex5 (aun (apow a2) (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → aal5 (aun a4 (asing (asm (apow a2) a2)))
In Proofgold the corresponding term root is bd3e84... and proposition id is 8a32e3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMM26iuLGdqnxMzqbARvht2LGQtHXmajWBG)
ain a2 a4 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (¬ (a2 = a3)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a0 (asm (apow a2) (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm (apow a2) (apow (asing a1))) → ain a2 (asm (apow a2) a2) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (asm a3 (asing a1) = a3)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing a2) a4) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a0 (asm a4 a2)) → ain a3 (asm a4 (asing a1)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a1 (aun a2 (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq a3 (asm a4 (asing a3)) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (apow a2) → aal4 a4 → aex2 (asm a4 a2) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun a3 (asing (apow a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain a1 (apow (asm a3 (asing a1)))
In Proofgold the corresponding term root is 0205fb... and proposition id is b7dfb0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLsAqLAtHXvGat5Earp7Y3SJWvmHMbwabV)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a1 a3 → ain a2 a3 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (¬ (a1 = a3)) → ain a0 (asm (apow a2) (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (asing a1 = apow a2)) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (apow (asing a1)) a3) → (¬ (a1 = apow a2)) → (¬ (asing (asing a1) = a3)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain a0 (asm a4 (asing a1)) → ain a1 (apow (asm a3 (asing a1))) → ain a0 (asm a4 (asing a2)) → (¬ ain (apow a2) a4) → ain a3 (asm a4 a2) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (asm a3 (asing a2) = a2) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing a2)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm a4 (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (apow a2) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))))
In Proofgold the corresponding term root is 7bd8c8... and proposition id is 57d7b9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGD9HAM3SUUAenU26nann7Y4PatSmDRePi)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a0 a3 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a1 (asm (apow a2) (apow (asing a2))) → asubq a1 (asm a3 (asing a1)) → (¬ asubq (asing a2) a2) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ asubq a3 (asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a2) (asing a3)) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun a3 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm (apow a2) a2) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2)))
In Proofgold the corresponding term root is 10abf0... and proposition id is b573a0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLaPCtEvhiY2YGBfzbVc2vaRn8RHq2zEC5)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a2 a1) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (asing a1) → (¬ (a0 = asm (apow a2) a2)) → ain (asing a1) (apow (asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) (asing a2)) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = apow a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ asubq a3 (asing a2)) → (¬ ain (asm (apow a2) a2) a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (a3 = apow a2)) → (¬ ain a1 (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain a2 (asing a3)) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a3 (asm a4 (asing a2)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (asm (apow a2) (asing a1)) → aal4 (apow a2) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun a3 (asing (apow a2))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (¬ ain a3 (apow (asing a2)))
In Proofgold the corresponding term root is f10557... and proposition id is 0e4c0d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUFNmAM83Jpb9QVXfVxtWPYJSoAuQwXPX1)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a0) → (¬ ain a2 a2) → (¬ asubq a4 a3) → ain a1 (apow a2) → (¬ ain a1 (apow (asing a1))) → (¬ (a0 = asing (asing a1))) → (¬ (a0 = asm a3 (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (a1 = asing (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ (a2 = apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (asing a2 = a3)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ (a3 = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a0 (asm a4 (asing a1)) → ain a0 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain (asing a2) (asing a3)) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a4 (aun a4 (asing (apow a2))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2)
In Proofgold the corresponding term root is 449012... and proposition id is dee7ea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJD99c2Vrrf4LznK9U5wySa39NiTeATxDk)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a3 = apow a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain a1 (apow (apow (asing a1))) → ain (asing a2) (apow (asing a2)) → (¬ ain (asm (apow a2) a2) a4) → ain a1 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm a2 (asing a1)) a1 → (asm a3 (asing a2) = a2) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (asm a3 (asing a1)) a4 → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4)
In Proofgold the corresponding term root is 1cb9c4... and proposition id is 3b2d0d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMaxWnSF4HReGBkn9v4sPLF2Mh8hoLFxXF6)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → asubq a1 a2 → (¬ asubq a2 a1) → (¬ asubq a4 a3) → (¬ ain a1 (apow (asing a1))) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ asubq a2 (asing a2)) → (¬ asubq (asing a2) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ (a3 = apow a2)) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a1 (apow (asm a3 (asing a1))) → ain a0 (asm a4 (asing a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal2 (asing a2)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))
In Proofgold the corresponding term root is 16f9b8... and proposition id is 97503e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWLCpj6tMJ93BP9KAm1GfNBxjybCiJPjdR)
(∀X24 : set, (¬ ain X24 a0)) → ain a0 a2 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a0 (asm a3 (asing a1)) → (¬ asubq a1 (asing a2)) → (¬ ain (asing a2) a1) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asing (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (asing a1 = apow a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (apow (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (asing a2) a4) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain a1 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asing a1) (asm a2 (asing a0)) → (asm a2 (asing a0) = asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq a2 (asm a3 (asing a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (asm a4 (asing a3)) a3 → asubq (asm a3 (asing a1)) a4 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm a4 (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))
In Proofgold the corresponding term root is b5bc11... and proposition id is 84173b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPVCAcixBkzEoBvGX7Y5fGzq3PQepBPZw6)
(∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (¬ asubq a2 a1) → (¬ (a1 = a2)) → ain a1 (asm (apow a2) (apow (asing a2))) → asubq a1 (apow (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ ain (asing (asing a1)) a3) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a3 (asm a4 (asing a2)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → ain a0 (aun a2 (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (asm a2 (asing a1)) a1 → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (asm a4 a2) → aal5 (aun a4 (asing (apow a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (apow (asing a1)) → aex2 (asm a4 a2) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0))
In Proofgold the corresponding term root is 1f68ed... and proposition id is accd3a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPGmVsbx8oE5Ng6bebLdTHgWq7mUhrbMNb)
ain a0 a2 → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (¬ (a1 = a3)) → ain a0 (asm (apow a2) (asing a2)) → (¬ asubq a1 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (apow a2) → ain (asing a1) (apow (asing a1)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (asing a1)) → (¬ (a1 = asing a2)) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (apow a2) (apow a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ (asing (asing a1) = a3)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain a0 (asm a4 a2)) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a3 (asing a2) = a2) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun a1 (asing a3)) → aex3 (asm (apow a2) (asing a0)) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1)))
In Proofgold the corresponding term root is 160fd8... and proposition id is 5635c2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRweBhg7BpMDRVnVxBsYzRiK2YvYxfEdiw)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (¬ ain a2 a2) → (¬ (a1 = asing a1)) → (¬ ain (asing a2) a1) → (¬ asubq (asing a2) a2) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (a3 = asm (apow a2) a2)) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asing a2) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (¬ aal2 (asing a1)) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 (asm (apow a2) a2) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2)
In Proofgold the corresponding term root is 7504a1... and proposition id is 1fd0d7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMT4ZfJzGMq4BVYSZDMAcoZRV7fT9aBWKYL)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → ain a0 a2 → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain (asing a2) a3) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asing a2 = a3)) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a0 (asm a4 a2)) → (¬ ain (asing a1) (asm a4 a2)) → ain a3 (asm a4 (apow (asing a2))) → ain a1 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a0 (aun a2 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a4 (aun (asing (asing a1)) a4) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing (apow a2))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal2 (asm a4 a2) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (asm (apow a2) a2) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2)))
In Proofgold the corresponding term root is f94530... and proposition id is 7e3fdb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPFXxHcme9FhDUqs3DBGUxh8pXPqqQ6xeo)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a1 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ asubq a1 (asing a1)) → ain (asing a1) (apow a2) → (¬ ain (asing a1) a1) → (¬ ain a2 (apow (asing a2))) → (¬ (a1 = asm a3 (asing a1))) → (¬ (asing a1 = apow a2)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (asing (asing a1) = a3)) → (¬ (asing a2 = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a2 (aun a1 (asing a3))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a3 (asm a4 a2) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a3 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a2 (asm a3 (asing a2)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm a4 (asing a3)) a3 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal5 a4) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aun a2 (asing (apow a2))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (apow a2) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ (asm (apow a2) a2 = apow a2))
In Proofgold the corresponding term root is 305cb2... and proposition id is 052c55...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRz77dQqYQxqVEjTypfbvs4TxapKd9Poks)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a2 a3 → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → ain a2 (asing a2) → (¬ ain a1 (apow (asing a2))) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm a3 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ ain (asing a2) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asing a2) (apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (apow a2) a3) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (apow (asing a1) = a3)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal4 a3) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal4 a4 → aex3 (asm a4 (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))
In Proofgold the corresponding term root is 5d0797... and proposition id is 178f75...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbHUsRMhmTRdVYytT169WDy7CfhCtyoZ6t)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ (a0 = a1)) → ain a0 (apow a2) → ain (asing a1) (asm (apow a2) a2) → (¬ ain (asing a2) a1) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain (asing a2) a2) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → ain a0 (apow (apow (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain a3 (apow (asing a2))) → ain a3 (asm a4 (apow (asing a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal2 (asm a4 a2) → aex2 a2 → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex4 (apow a2) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (asing a2)) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is f3eced... and proposition id is 579841...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMaMrYqePqp7Ha1J8oSMjEdnqMx4x2djg4G)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (¬ ain a1 a1) → (¬ asubq a2 a1) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (a1 = asing (asing a1))) → (¬ ain (apow a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (apow (asing a1)) a3) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a0 (asm a4 (asing a2)) → (¬ ain (asm a3 (asing a1)) a4) → ain a3 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (asm a3 (asing a2) = a2) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 (asm a4 a2)) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3)))
In Proofgold the corresponding term root is e50470... and proposition id is 83b76b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTsxSfRxGWjVRRTeEi7P1trebRkd6Nhyz3)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a1 a0) → (¬ (a1 = a2)) → (¬ ain (asing a1) a2) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (asing (asing a1) = a3)) → ain a0 (apow (asing (asing a1))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (asing (apow a2)) → ain a0 (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a2 (asing a0) = asing a1) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (asm a4 (asing a3) = a3) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (apow a2)) → aal3 (aun (asing a2) (apow (asing a2))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (asm (apow a2) (asing a0)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))))
In Proofgold the corresponding term root is b1e7ed... and proposition id is 263f71...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTdjXMq5VMTs1FS6fUAWFMz2vGVY2w6ERY)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a3 a2) → (¬ asubq a3 a2) → ain a1 (apow a2) → (¬ (a0 = asing (asing a1))) → ain a2 (asm (apow a2) (asing a1)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a1) (asing a2)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) a3) → (¬ (a0 = apow a2)) → (¬ (asing a2 = apow a2)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain a0 (asm a4 (asing a2)) → (¬ ain (asm (apow a2) a2) a4) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a0 (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a3)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (apow (asing (asing a1))) → aal3 (aun a2 (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 a3 → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow X24))
In Proofgold the corresponding term root is 10553d... and proposition id is d188da...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMaKwSDTeb9hHYQdjbKW2SKuJrgKEYQAqtY)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ asubq a4 a3) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a0 (apow (asing a1)) → (¬ ain a0 (asing a2)) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = asm a3 (asing a1))) → ain a2 (apow a2) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) a2) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a1 = apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a0 (aun a3 (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → aal2 (aun (asing a1) (asing (asing a1))) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))
In Proofgold the corresponding term root is dcb5a2... and proposition id is b2bb4f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJnvL3kV7buED66VrLdYnNwi5vCDrZSxr1)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → asubq a1 a2 → (¬ asubq a1 a0) → asubq (asing a1) a2 → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (apow a2) (asing a2)) → (¬ asubq (apow a2) (asing a2)) → asubq a3 (apow a2) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a2 (asing a3)) → (¬ ain (asing a2) (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (asm a3 (asing a2) = a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 a4) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal4 a4 → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1))
In Proofgold the corresponding term root is fad092... and proposition id is da2261...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMK1FBqxBC99GWZBgYzfzDSmCXJHrtjpZfJ)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ (a2 = a3)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a1 (asing a3)) → (¬ ain (asing a2) (asing a3)) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow (asing a1)) a4) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain a0 (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a4 (aun a4 (asing (apow a2))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm a4 a2) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2)))
In Proofgold the corresponding term root is 2b2be7... and proposition id is bb49cf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMm2h8q32tHa5cHYgjtkEaoortCRUcqntP)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (¬ asubq a4 a3) → (¬ ain (asing a1) a0) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ (a0 = asing a1)) → (¬ (asing a1 = a2)) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ ain (apow (asing a1)) a3) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow (asing a1)) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) a4) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asing a2) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) a4 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (apow (asing a2)) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a2)) → aex5 (aun (asing (asing (asing a1))) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ ain a3 (aun (apow a2) (asing (apow a2))))
In Proofgold the corresponding term root is 050e8e... and proposition id is 0bf5a6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGvrpZ9UpCa59yBhvL95PPEmwxmW3359xS)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → ain a0 (asm a3 (asing a1)) → (¬ asubq a1 (asing a2)) → ain a2 (asm a3 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a1))) → ain a0 (apow (asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = apow a2)) → (¬ ain (apow a2) (asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ (asing (asing a1) = a3)) → (¬ (asing a2 = a3)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (apow a2) (asing (asing a1)) = a3) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal2 (asing a3)) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (asm a4 (asing a2)) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing a2)) a4) → aal5 (aun a4 (asing (apow a2))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (aun (asing (asing a2)) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1))))
In Proofgold the corresponding term root is b6380a... and proposition id is b33eee...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcF7945S1fMDQMPdHR9M6gNYqtnS77j5kT)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (¬ ain a3 a3) → ain a0 (apow (asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (asing a1 = asing a2)) → (¬ (asing a1 = a3)) → (¬ ain (apow a2) (apow a2)) → (¬ asubq a3 (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (apow (asing a1)) a3) → (¬ (a1 = asm (apow a2) a2)) → (¬ (a1 = apow a2)) → (¬ ain a0 (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain a2 (asing a3)) → ain a3 (aun (asing a1) (asing a3)) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a0 (aun a1 (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal2 a1) → (¬ aal2 (asing a3)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 a2 → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aun (asing (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex5 (aun (apow a2) (asing (apow a2))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ (a0 = apow (asing a1)))
In Proofgold the corresponding term root is 77b7f5... and proposition id is 8a68ea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMREmrjzaNvKzJHXaVzUavg9kUX8E61daGH)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ ain (asing a1) a1) → (¬ (a0 = asing (asing a1))) → (¬ asubq a2 (asing a1)) → ain a2 (asm (apow a2) a2) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) a2) a3) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (apow (asing a1) = a3)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain a3 (asing (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a0 (asm a4 a2)) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun a1 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (asm a2 (asing a0)) (asing a1) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (¬ aal2 (asing (asing a1))) → (¬ aal2 (asing a2)) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal4 a4 → aal4 (aun a3 (asing (apow a2))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 a3 → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm a4 (asing a3)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a2)) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a1 → (X24 = a0))
In Proofgold the corresponding term root is 5b8292... and proposition id is 06be58...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSTtuex99bGR56LWNYnpRZYPxFqcJDGzLk)
ain a0 a1 → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → asubq a3 a4 → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain a3 (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ (a2 = asing (asing a1))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (asing a1) (asm a4 a2)) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a2 (aun a3 (asing (apow a2))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal2 (apow (asing (asing a1))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm a3 (asing a1)) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm a4 (asing a3)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is dcd80e... and proposition id is 123f9c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMX3V7t6SU8Yg8D5tD1XbUS2TGJFyBHXDZN)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a2 a3 → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → asubq a3 a4 → (¬ (a0 = a3)) → ain a1 (asing a1) → ain a0 (apow a2) → (¬ ain a1 (apow (asing a2))) → (¬ ain (asing a2) a1) → (¬ (a0 = asm a3 (asing a1))) → (¬ asubq a2 (asing a1)) → (¬ ain (asing a2) a2) → (¬ (asing a1 = asing a2)) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → ain a0 (apow (apow (asing a1))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a0 (asing a3)) → (¬ ain (asing a2) (asing a3)) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow (asing a1)) a4) → (¬ ain a0 (asm a4 a2)) → ain a1 (aun (asing (asing a2)) a4) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (¬ aal2 (asing a3)) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex5 (aun (apow a2) (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (apow a2) (asing (asing a2)))
In Proofgold the corresponding term root is 84f1ba... and proposition id is aa81db...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVVFTHeQYPFuzANkbPqbfWYcAq5kH39y35)
(∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a4 a3) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (asing a1) → ain a0 (apow (asing a1)) → (¬ ain a1 (apow (asing a1))) → ain (asing a1) (apow a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ (asing a1 = a2)) → asubq a1 (asm a3 (asing a1)) → (¬ (a1 = asm a3 (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ (a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain a3 (asing (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a1 (aun (asing (asing a2)) a4) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → ain a2 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (apow a2))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (asing (apow (asing a1))) a4) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (apow (asing a1)) → aex2 (asm a3 (asing a1)) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ asubq (aun a4 (asing (apow a2))) a4)
In Proofgold the corresponding term root is 05356e... and proposition id is 899f1e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLXc2oC7zS5A74ZCDyZRqhNegQ2Hg42LAm)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → asubq a1 (apow (asing a1)) → (¬ asubq (asing a2) a2) → (¬ (apow (asing a1) = a3)) → asubq a3 (apow a2) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a2 (asing a3)) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → ain a3 (asm a4 (asing a1)) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a2)) a4) → (¬ ain a0 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a2 (asm a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq a2 (apow (asm a3 (asing a1))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal3 (aun a2 (asing (apow a2))) → aex3 (asm a4 (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1)))))
In Proofgold the corresponding term root is b7a9e0... and proposition id is 90e07c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMG8JGkYxGGzPjZK67xjkBnYQgWq6BpY1Js)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → ain a1 (asing a1) → (¬ ain (asing a2) a1) → ain a2 (apow a2) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (asing a1)) → (¬ (a1 = asing (asing a1))) → (¬ ain (asing a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ (asm a3 (asing a1) = a3)) → (¬ ain a0 (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a0 (asm a4 a2)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a0 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 a3 → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24))))
In Proofgold the corresponding term root is 7d75f1... and proposition id is 1097c9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMEp1GwEYeyjBfANhJenMnPvWDuDs4Zhdp3)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a3 a2) → (¬ (a0 = a1)) → ain a0 (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (asing a1 = asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ (asing (asing a1) = a3)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ (a3 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asm (apow a2) a2 = apow a2)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a2)) → ain a2 (asm a4 a2) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a3 (asing a2) = a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing a2)) a4) → aex3 (aun a2 (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ aal3 (aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is 6af031... and proposition id is 7bb982...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMY9qdCzDoiMqW7p6iZL43djwWkQAt4ZSXa)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a0 a4 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ asubq a2 a0) → ain a0 (apow a2) → (¬ (a0 = asing a2)) → (¬ (asing a1 = a2)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ (asing a1 = a3)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → (¬ ain a1 (asing a3)) → (¬ ain (asing a2) (asing a3)) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (apow a2) (asing (asing a1))) a3 → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm a4 (asing a3)) a3 → asubq (asm a3 (asing a1)) a4 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (apow (asing a2)) → aex2 (aun a1 (asing (apow a2))) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (apow a2) (asing (apow a2))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2)
In Proofgold the corresponding term root is adf06e... and proposition id is 0c96b4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQDjoaAYEXWrACgT8d5aRTMAWYLRyjCCnt)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a1 a3 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → ain a0 (apow a2) → (¬ (a0 = asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing (asing a1)) a3) → (¬ (a0 = apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a2 (asm a4 a2) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a3 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 a3 → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 a4 → aex2 (asm a3 (asing a1)) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1))
In Proofgold the corresponding term root is fac56c... and proposition id is 2f4250...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPvAoWGwuaBCccUQASyRzGah44dEhRVgAA)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → asubq a3 a4 → ain a1 (apow a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain (asing a1) a1) → asubq a1 (apow (asing a1)) → (¬ ain (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (a1 = apow (asing a1))) → (¬ ain (apow a2) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain (apow (asing a1)) a4) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a3 (aun a3 (asing (asing a2))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 (asm (apow a2) a2) → aal3 (asm (apow a2) (apow (asing a2))) → aex2 (aun a1 (asing (apow a2))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1))))
In Proofgold the corresponding term root is 1ec2cf... and proposition id is e1444d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTgSPceuSag58ANxBm6xRPmBKfaXMwhxiD)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (¬ ain a0 a0) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = asm (apow a2) a2)) → (¬ (a1 = asm a3 (asing a1))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → asubq a3 (apow a2) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (a3 = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (asing a1) a4) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a1 (asing a3)) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (asm a2 (asing a0)) (asing a1) → (asm a2 (asing a1) = a1) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 a3 → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun a3 (asing (apow a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain a0 (apow X24))
In Proofgold the corresponding term root is ef6ff2... and proposition id is 5abd6a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWp6A99NZqFSaFtpx3vk27ivaLf2b8yijH)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a2 (asm a3 (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ ain a2 (apow (asing a2))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asing (asing a1))) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = a3)) → (¬ (asing a1 = asing (asing a1))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain a3 (apow a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a1 (asm a4 (asing a2)) → ain a3 (asm a4 a2) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a3 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 a2 → aal2 (asm (apow a2) (apow (asing a1))) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))
In Proofgold the corresponding term root is 4fba86... and proposition id is ffbb33...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNZY66tgprvH2ZRVi5rRtoUNh1KR6wZ2gj)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (¬ asubq a2 a1) → (¬ asubq a2 a0) → asubq (asing a1) a2 → ain a0 (asm (apow a2) (asing a1)) → (¬ ain (asing a1) a1) → (¬ ain (asing a2) a1) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (a2 = asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (apow (asing a1) = a3)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (asm a4 (apow (asing a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm a3 (asing a2)) a2 → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal3 (asm a4 (asing a1)) → aal4 (aun a3 (asing (asing a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (¬ ain a0 (aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is efadf4... and proposition id is 4961d7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPjiBbUTEdLNd5QPQ7RRFYg4B9s5dxAYx1)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a3 a3) → (¬ (a2 = a3)) → (¬ ain (asing a1) a0) → ain (asing a1) (asing (asing a1)) → ain a0 (apow (asing a1)) → (¬ ain a0 (asing a1)) → (¬ ain (asing a2) a1) → asubq a1 (asm a3 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ ain a2 (apow (asing a2))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain a2 (asing a3)) → ain a3 (aun a1 (asing a3)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain a1 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal2 (asm a4 a2) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (aun a2 (asing (apow a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun a4 (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex4 (aun a3 (asing (apow (asing a1)))) → aex3 (asm a4 (asing a0)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is f95683... and proposition id is bc6b5a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMd5FgbREUBe3vg3uqG9LvVgMzLGzH8aLMc)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ ain a1 a0) → (¬ ain a3 a3) → (¬ (a0 = a1)) → (¬ asubq a1 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ (a1 = asing a1)) → ain (asing a1) (asm (apow a2) a2) → (¬ asubq a1 (asing a2)) → (¬ (a0 = asm a3 (asing a1))) → ain a0 (apow (asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (asing a2 = a3)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a0 (asm a4 (asing a2)) → (¬ ain (asing (asing a1)) a4) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a2 (asm a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (asm a4 (asing a3) = a3) → asubq a2 (apow (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (¬ aal2 (asing (asing a1))) → (¬ aal3 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 (apow (asing (asing a1))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm a3 (asing a2)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex5 (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25))
In Proofgold the corresponding term root is d47fd4... and proposition id is eaf01e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdn2U1o7QXrLJFg6A89b5mr4AsV62MmVR7)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a0 a2 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a3 a2) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain (asing a1) (asing (asing a1)) → (¬ ain (asing a1) a1) → (¬ (a0 = asing a2)) → (¬ ain (asing a2) a1) → (¬ ain a2 (apow (asing a1))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (asing a1 = asing a2)) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = apow a2)) → (¬ (a2 = asing a2)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → ain a1 (apow (apow (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a2) a4) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal3 (asm (apow a2) (asing a2)) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex5 (aun a4 (asing (apow a2))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24))
In Proofgold the corresponding term root is f36e17... and proposition id is f19555...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHwTGCXes7r15G4NUhy9Ym59xMjZXFEvsf)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a0 a0) → (¬ ain a2 a1) → (¬ (a0 = a2)) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ asubq a1 (asing a2)) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a3 (asing a1)) → (¬ asubq (asing a2) a2) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ (a2 = asing a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → aal3 (asm (apow a2) (apow (asing a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (apow (asing a1)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (asing (apow (asing a1))) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2)))
In Proofgold the corresponding term root is 63c34b... and proposition id is 47290d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLxHMznEqc1NP5L7pSUjmQHEfv3BTsUcnE)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a1 a4 → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ (a0 = a1)) → ain a0 (apow (asing a1)) → asubq (asing a1) a2 → ain a1 (asm (apow a2) (asing a2)) → (¬ (a0 = asing a2)) → asubq a1 (apow (asing a1)) → ain a2 (asm a3 (asing a1)) → ain a2 (apow a2) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asing a1) a3) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (apow a2) (apow a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ (asing a2 = apow a2)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (apow (asing a1)) a4) → (¬ ain (asm (apow a2) a2) a4) → ain a1 (asm a4 (apow (asing a2))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) a4 → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal3 (asm a4 (asing a1)) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (apow (asing a2)) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain a0 X24 → asubq a1 X24)
In Proofgold the corresponding term root is eeddcf... and proposition id is 661f74...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMEybGh3AXCS89Exii5sCganGHNCVeze9oz)
(∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (¬ ain a2 a2) → (¬ (a2 = a3)) → ain a1 (asing a1) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ (a1 = asing a1)) → ain a0 (asm (apow a2) (asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (asing a1 = a2)) → ain a2 (asm a3 (asing a1)) → (¬ ain a3 (asing a1)) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = a3)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (apow a2) (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain a0 (asing (apow (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a0 (asm a4 (asing a2)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a0 (aun a1 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal3 (asm a4 (asing a1)) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal5 (aun a4 (asing (apow a2))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → ain a0 (apow (asm a3 (asing a1)))
In Proofgold the corresponding term root is 43066c... and proposition id is 4834bb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKS6vwXMmca9E18S85xaNCZCfcZi7BTwsG)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ asubq a2 a0) → ain a0 (asm a3 (asing a1)) → (¬ ain (asing a1) a2) → (¬ ain a0 (asing (asing a1))) → ain (asing a1) (apow (asing a1)) → (¬ ain (asing a1) (asing a2)) → ain a2 (apow a2) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (asing a2) a2) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a2 = asing a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain a3 (aun a1 (asing a3)) → ain a3 (asm a4 (asing a2)) → ain a3 (asm a4 (asing a1)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal4 (aun a3 (asing (apow a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm a4 a2) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun (apow a2) (asing (apow a2))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25)))
In Proofgold the corresponding term root is d1b204... and proposition id is 26fa93...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMEy67PcKjexGYKjjuWgh6eWftsU9ntFL98)
ain a0 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → ain a2 (asm a3 (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ (asing a1 = asing a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (asing a1)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a0 (aun a2 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal2 a1) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 a3 → aal4 (apow a2) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm a4 (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 a4 → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (¬ ain (asing (asing a1)) a2)
In Proofgold the corresponding term root is a47c17... and proposition id is bddf91...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYVGoE4v3JrBrbAGyFQ148LYXTvYQMFY8G)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a2 a4 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a1 a0) → ain (asing a1) (asm (apow a2) a2) → (¬ asubq a2 (asing a1)) → (¬ ain a2 (apow (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ asubq (asing a2) a2) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (apow (apow (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain a2 (aun a1 (asing a3))) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (asm (apow a2) (asing a2)) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aex2 (apow (asing a1)) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun a1 (asing (apow a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex5 (aun (asing (asing (asing a1))) a4) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25))
In Proofgold the corresponding term root is f99b47... and proposition id is 8e5bad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUMjAnrrBR6jsfNZttBbrtF78wtAGWVRNR)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a0 a0) → (¬ asubq a2 a1) → asubq a3 a4 → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a0 (apow a2) → (¬ ain a1 (apow (asing a1))) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a2 (apow (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asing a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain a0 (asm a4 a2)) → ain a3 (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a0 (aun a4 (asing (apow a2))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (asm (apow a2) a2) → aex3 (asm a4 (asing a2)) → aex4 (apow a2) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq a1 (asm a3 (asing a1))
In Proofgold the corresponding term root is d8ee8a... and proposition id is eaab94...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRsBed3Qkwt9fNEMqKRqvRQw8UVj45jtDN)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → ain a0 a3 → ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (¬ (a1 = a2)) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain (asing a1) (apow a2) → ain (asing a1) (asm (apow a2) a2) → (¬ ain (asing a1) a1) → ain (asing a1) (apow (asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (asing a1 = a3)) → (¬ (asing a1 = apow a2)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ (asing a2 = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a0 (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a3 (aun a4 (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal2 (asm (apow a2) a2) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex3 (asm a4 (asing a0)) → aex3 (asm a4 (asing a3)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (asing a2)) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2))
In Proofgold the corresponding term root is 7b832f... and proposition id is c52fb4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRcHRx1iKq9TwzKMixuJfCUFaTSYWQY12w)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (¬ (a1 = a2)) → (¬ (a0 = asm (apow a2) a2)) → asubq a1 (apow (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a2 = asing a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (a3 = apow a2)) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a1 (asing a3)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a3 (asing a2) = a2) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ asubq (aun a3 (asing (apow a2))) a3)
In Proofgold the corresponding term root is 466960... and proposition id is 6b3502...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMcTMNWUGLMkZQV46Kt7VWoZCcWzfGoXZM)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a1 a0) → (¬ ain a2 a2) → (¬ asubq a3 a2) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (¬ ain (asing a1) a0) → (¬ ain (asing a1) a1) → (¬ ain a1 (asing a2)) → (¬ ain (asing a2) a1) → (¬ ain a2 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (asing a1) a3) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a1 = apow a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (asm (apow a2) a2 = apow a2)) → ain a1 (apow (apow (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) a4) → ain a2 (asm a4 (asing a1)) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → (¬ ain a3 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 a4 → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aun a1 (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0))
In Proofgold the corresponding term root is 34bae3... and proposition id is 508641...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYBNyykM6ezXCqwAHsHYejRitifCLdrPn3)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a2 a0) → ain (asing a1) (asing (asing a1)) → (¬ ain (asing a1) a1) → asubq a1 (apow (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a1 (asm a4 (asing a2)) → (¬ ain a0 (asm a4 a2)) → ain a2 (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (apow a2) (asing (apow a2)) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm a3 (asing a2)) a2 → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 a3) → aal2 (asm (apow a2) a2) → aex3 (aun a2 (asing (apow a2))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing a2)) a4) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25)))
In Proofgold the corresponding term root is 3b46eb... and proposition id is 8d7c64...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHmayAE1o4DMJZgsQfnDSHSpre5zwc4jHf)
ain a0 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a1 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a0 (asm a3 (asing a1)) → (¬ ain (asing a2) a1) → (¬ (a0 = asm (apow a2) a2)) → ain a2 (asm (apow a2) a2) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asing a2) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (asing a1 = apow a2)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a1 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a0 (aun a4 (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → asubq a3 (aun a3 (asing (asing a2))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun a4 (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a4 (asing a3)) a3 → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 a2) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 (asm a3 (asing a1)) → aal3 (aun a2 (asing (apow a2))) → aal5 (aun (asing (asing a1)) a4) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm a3 (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm a4 (asing a3)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))))
In Proofgold the corresponding term root is 3665e7... and proposition id is 2157aa...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFiio4fdDAnjxK64UZWaRNkGH4Vpb3qXQo)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a1 a2 → ain a1 a4 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a2 a1) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a2 (asing a2) → (¬ asubq a1 (asing a1)) → asubq (asing a1) a2 → (¬ ain a1 (asing a2)) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain a1 (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (apow (asing a1)) a3) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (a3 = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain a0 (asm a4 a2)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm a4 a2)) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 (asm a4 a2) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun a1 (asing a3)) → aex2 (asm a4 a2) → aex2 (aun a1 (asing (apow a2))) → aex3 (asm (apow a2) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2)
In Proofgold the corresponding term root is 3f37e3... and proposition id is cedde5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMuU6kuwQPcydizv46uFXJm94AmrhcHZhG)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (¬ ain a1 a0) → (¬ ain a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (asm (apow a2) (asing a2)) → (¬ (a0 = asing a2)) → asubq a1 (asm a3 (asing a1)) → (¬ asubq a2 (asing a2)) → (¬ (a1 = asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asing a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (a1 = apow a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (apow (asing a1)) a4) → ain a2 (asm a4 a2) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25))
In Proofgold the corresponding term root is 993dc4... and proposition id is 0cbcff...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMF4Q9r9X7bVNmn5CWE9KzZKRDVFY2r2PrN)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a3 a2) → (¬ asubq a2 a1) → (¬ (a0 = asing a2)) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (asing a1 = apow a2)) → (¬ ain (asm (apow a2) a2) a3) → (¬ (a1 = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a3 (aun a1 (asing a3)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain a1 (asing (apow a2))) → (¬ ain a3 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (asm a4 (asing a3)) a3 → (asm a4 (asing a3) = a3) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (apow a2)) → aal2 a2 → aal4 a4 → aex2 a2 → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (asm a4 (asing a3)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a3 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 72437a... and proposition id is aee213...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTjbZ2nxCGEVvQniCnxivEMtoKXN7QWg2P)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ asubq a1 a0) → (¬ ain a0 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a1))) → ain a2 (asm (apow a2) a2) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (a1 = apow a2)) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → ain a1 (aun a3 (asing (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a0 (asing a3)) → ain a3 (aun a1 (asing a3)) → (¬ ain a0 (asm a4 a2)) → ain (apow a2) (aun a1 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal5 (aun (asing (asing a1)) a4) → aal5 (aun (asing (asing a2)) a4) → aal5 (aun a4 (asing (apow a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm a3 (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))))
In Proofgold the corresponding term root is 53a9a7... and proposition id is c6a281...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcHwhoHVuVmakuG5xfLhnAcgxMG9Vg33YQ)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ (a0 = a2)) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ (a1 = asm (apow a2) a2)) → asubq a3 (apow a2) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) a4) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain a0 (asing a3)) → ain a3 (asing a3) → (¬ ain (apow a2) a4) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a1 (asm a2 (asing a1)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 (asm (apow a2) a2) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (asm a4 (asing a2)) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → asubq a2 (aun a2 (asing (apow a2)))
In Proofgold the corresponding term root is eca9d7... and proposition id is ede9f0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLAQhNga7CjwzE2k9sWXeSMw8BRmH4aqnX)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain a0 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = apow (asing a1))) → (¬ asubq a2 (asing a1)) → (¬ ain a2 (apow (asing a2))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) a2) a3) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (asm a3 (asing a1) = a3)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a2 (asing a3)) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain (asing a2) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 a2) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal2 (asm (apow a2) a2) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm a4 (asing a2)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex5 (aun a4 (asing (apow a2))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))
In Proofgold the corresponding term root is 803a34... and proposition id is 562c8a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSe4LDPbSJYNQkhDVgZr6jMYmMA57q5EgL)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → ain a0 a4 → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a2 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain (asing a1) (asm (apow a2) a2) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a0 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asing a1) (asm a2 (asing a0)) → (asm a2 (asing a1) = a1) → asubq (asm a3 (asing a2)) a2 → (asm (asm a3 (asing a1)) (asing a2) = a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 a1) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal4 a4 → aal5 (aun (asing (asing (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aun a1 (asing (apow a2))) → aex3 a3 → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → ain a0 (apow (apow (asing a1)))
In Proofgold the corresponding term root is 9766b0... and proposition id is 5b454f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWv8PF2EdPaYF5i3eb2AwrNzJymQpksmAS)
(∀X24 : set, (¬ ain X24 X24)) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ (a0 = apow (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq a2 (asing a1)) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) a3) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a0 (aun a1 (asing a3)) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal2 (apow (asing (asing a1))) → aal3 a3 → aal5 (aun (asing (asing a1)) a4) → aex2 (aun a1 (asing (apow a2))) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex5 (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))))
In Proofgold the corresponding term root is 4e5aba... and proposition id is 2060c1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWJUswiZrRWGRynwg53x1F2KiaRdLoHvEE)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → asubq a1 (asing a0) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → asubq a2 a3 → asubq a3 a4 → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (¬ asubq a2 (asing a2)) → (¬ (a1 = asing a2)) → (¬ asubq (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain (asing a2) (asing a3)) → (¬ ain (apow (asing a1)) a4) → ain a2 (asm a4 (apow (asing a2))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a2) (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (asm a2 (asing a0)) (asing a1) → (asm a2 (asing a1) = a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal2 (asing (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a2))) → aal3 (asm a4 (asing a2)) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (asing (asing a1)) a4) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (apow a2) (asing (apow a2))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2))
In Proofgold the corresponding term root is 0cdb43... and proposition id is ba12b3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdnM56nd17ZK7snWkZYti3w8CsVdKVGsd1)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → asubq a1 a2 → asubq a3 a4 → (¬ (a0 = a2)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (¬ ain a0 (asing a2)) → (¬ ain (asing a1) a1) → ain a2 (asm (apow a2) (apow (asing a1))) → ain a2 (asm (apow a2) (apow (asing a2))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a2) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ (a0 = apow a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a3 (asing a3) → ain a0 (aun a1 (asing a3)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a3 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 a3 → aal5 (aun (apow a2) (asing (apow a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1))))
In Proofgold the corresponding term root is 2f60e9... and proposition id is 447115...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMb1fXoHD1gVBFLxp4dZV5o8yVzpJYB3CR9)
ain a0 a2 → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a0 a0) → ain a0 (asm a3 (asing a1)) → ain a1 (apow a2) → ain (asing a1) (apow a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = apow (asing a1))) → (¬ ain a0 (asing (asing a1))) → ain a2 (asm (apow a2) a2) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain a3 (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (a3 = apow a2)) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a1)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq a3 (asm a4 (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun (asing a3) (asing (apow a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun a3 (asing (asing a2))) → aex3 (asm a4 (asing a3)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ (asm a3 (asing a1) = apow a2))
In Proofgold the corresponding term root is 9507ae... and proposition id is 2025f5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMd3yjb8S3jarny4wVpJmUqMDuhVYzbtRdG)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (¬ ain a2 a2) → (¬ (a0 = a1)) → (¬ (a2 = a3)) → ain a1 (asing a1) → ain a0 (apow a2) → (¬ ain a0 (asing a1)) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing a2) a3) → (¬ ain a3 (asm a3 (asing a1))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing a2)) a4) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a2 (asing a1) = a1) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 a4) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (asm (apow a2) (asing a1)) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (asm a4 (asing a1)) → aal5 (aun (asing (asing (asing a1))) a4) → aex3 (asm (apow a2) (asing a1)) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (apow (asing a1))) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ asubq (asm (apow a2) a2) a2)
In Proofgold the corresponding term root is 752701... and proposition id is 8bb4ca...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUyM9kNvfcoHDS2arr4TCxoW47b8jL51Q1)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a1 a1) → (¬ ain a2 a2) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (¬ asubq a1 (asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (apow (asing a1)) a3) → (¬ (a1 = asm (apow a2) a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a3 (asing a3) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a2) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0))
In Proofgold the corresponding term root is 4817ac... and proposition id is 0cac28...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGYjRnBesrNqt6oVPrgAW1eWJzXbaTw2Tv)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ (a1 = a3)) → ain a0 (apow (asing a1)) → ain a2 (asing a2) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a2 (asing a1)) → (¬ asubq a2 (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a1 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ asubq (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq a1 (asm a2 (asing a1)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2))))
In Proofgold the corresponding term root is 228efd... and proposition id is a0e7d4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMF9Dc7CKJAFeLSfTfH4dQ9mMCgUfB67wc5)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ (a0 = a3)) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ ain a0 (asing a1)) → (¬ ain (asing a1) a2) → (¬ ain a0 (asing (asing a1))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ (a2 = asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ ain a3 (apow a2)) → (¬ ain (apow a2) a3) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a2 (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → ain a2 (asm a4 (apow (asing a2))) → ain a3 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm a2 (asing a1)) a1 → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a3 (asing a2)) a2 → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal4 a4 → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (asm a4 a2) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25))
In Proofgold the corresponding term root is 3f5970... and proposition id is c7fa0c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMkJ7asVDEz8cHkpcz1n94KJx9QkQd7gWo)
(∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → ain (asing a1) (asing (asing a1)) → (¬ ain a0 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) a2) → (¬ (a0 = asing (asing a1))) → (¬ (a0 = asing a2)) → (¬ ain (asing a2) a1) → (¬ (a1 = asing a2)) → (¬ (a1 = asm a3 (asing a1))) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ asubq (apow a2) (asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a3 (aun a1 (asing a3)) → ain a3 (aun (asing a1) (asing a3)) → ain a0 (asm a4 (asing a2)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (asing (apow a2)) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a1 (asm a2 (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal3 (asm (apow a2) (asing a2)) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal5 (aun (asing (apow (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25)))
In Proofgold the corresponding term root is 1d90dd... and proposition id is 20da53...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdRU3oy1hytRsWSQQfCYPYQQkniFfCKVeH)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (¬ ain a2 a0) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain (asing a1) (apow a2) → ain a2 (asm (apow a2) a2) → (¬ ain (asing a2) a2) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (asing a1) a4) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → ain a0 (asm a4 (asing a2)) → (¬ ain a0 (asm a4 a2)) → ain a3 (asm a4 (asing a1)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (asing a2) (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (aun a2 (asing (apow a2))) → aal4 (aun a3 (asing (apow (asing a1)))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, aex2 (apow (asing X24)))
In Proofgold the corresponding term root is b7bcc1... and proposition id is e00b24...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYdnPFobGNo5Xi858gHhry1yab1KhwQFFw)
ain a1 a2 → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → ain a0 (apow (asing a1)) → asubq (asing a1) a2 → (¬ ain (asing a1) a1) → ain a1 (asm (apow a2) (asing a2)) → (¬ asubq a2 (asing a1)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (asing a1 = a3)) → (¬ (a2 = apow (asing a1))) → (¬ (a2 = apow a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = apow a2)) → ain a0 (apow (asing (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a1 (aun a3 (asing (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain (asing a2) (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm a4 (asing a3)) a3 → asubq a3 (asm a4 (asing a3)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal3 (asm (apow a2) (asing a1)) → aal4 (aun a3 (asing (apow (asing a1)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1))
In Proofgold the corresponding term root is bf12fe... and proposition id is 93d110...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSa5ZYGBzfZYeSdriy5Cub6E1zqHSyEjbq)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ ain a1 (apow (asing a2))) → ain a1 (asm (apow a2) (apow (asing a2))) → ain a2 (apow a2) → (¬ ain a1 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain a1 (apow (asm a3 (asing a1))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a2 (asm a4 a2) → ain a3 (asm a4 a2) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (apow (asing a1))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aun a2 (asing (apow a2))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ ain (asing a2) a4)
In Proofgold the corresponding term root is 77c1d3... and proposition id is 62e724...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKn7Qt9ShjXtBwG8VVtw5Rf8LAu2WPPYo6)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a2 a3 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a1 a0) → (¬ ain a0 (asing a2)) → ain a0 (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ (a1 = apow (asing a1))) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (apow (asing a1) = a3)) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain a2 (asing a3)) → (¬ ain (asing a2) (asing a3)) → ain a3 (asm a4 a2) → ain a3 (asm a4 (asing a1)) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain a3 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a4 (aun (asing (asing a2)) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a2 (asing a1)) a1 → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 a1) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun a3 (asing (asing a2))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2))))
In Proofgold the corresponding term root is 1feb74... and proposition id is e637ea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMBe6gubUUpnSY2JHQxVJS69TYwatvAdCQ)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a2 a0) → (¬ (a0 = a2)) → (¬ (a0 = asing (asing a1))) → ain a1 (asm (apow a2) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asing (asing a1))) → (¬ asubq (asing a1) (asing (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (apow a2) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (a0 = apow a2)) → (¬ (asing a2 = a3)) → asubq a3 (apow a2) → (¬ (asm (apow a2) a2 = apow a2)) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a2 (aun a1 (asing a3))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain a1 (aun a3 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun a3 (asing (apow a2))) → aex2 (asm (apow a2) a2) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun (apow a2) (asing (apow a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))))
In Proofgold the corresponding term root is 051d14... and proposition id is 6f6054...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMT14JPFh3x5T6jNm3tU1Ro1RX6NsSHe56o)
(∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a1) → (¬ asubq a1 a0) → asubq (asing a1) a2 → (¬ asubq a2 (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asing a2) a2) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain a3 (apow (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a3 (aun a1 (asing a3)) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal5 a4) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (apow a2))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)))
In Proofgold the corresponding term root is 66a6ff... and proposition id is 85ca87...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRuzRhbhJJjkb2vye4NUhLuYg98SunQy68)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (¬ (a0 = a2)) → ain a1 (apow a2) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ (a2 = apow (asing a1))) → (¬ ain (apow a2) (apow a2)) → (¬ (a1 = apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = apow a2)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain a2 (asing a3)) → (¬ ain (asing a2) (asing a3)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a2 (aun (asing (asing a2)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun a1 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → aal2 (asm (apow a2) a2) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (¬ ain a3 (aun (asing a1) (apow (asing a2))))
In Proofgold the corresponding term root is c8792e... and proposition id is dae538...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLN7WXoN9iMwEL6u4o1UF8NUXAhc4csqZM)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (¬ ain a3 a3) → (¬ (a0 = a2)) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = apow a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain a2 (asing a3)) → ain a1 (asm a4 (asing a2)) → ain a3 (asm a4 (asing a1)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (apow a2))) → (¬ aal5 a4) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (asm a4 a2) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm a4 (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ ain a1 a0)
In Proofgold the corresponding term root is 7580d1... and proposition id is 7f72e4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMTXLniz7C1YmAiqf4bEk6p3TRPrDp4Gx6)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (¬ ain a2 a0) → ain a1 (apow a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (apow (asing a1)) a3) → (¬ ain a3 (asm a3 (asing a1))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a3 (asm a4 (asing a2)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a3 (aun a3 (asing (asing a2))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (asing a1) (asm a2 (asing a0)) → (asm a2 (asing a0) = asing a1) → (asm a3 (asing a2) = a2) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (apow (asing (asing a1))) → aal3 (asm a4 (asing a2)) → aex2 (asm a3 (asing a1)) → aex2 (aun a1 (asing (apow a2))) → aex3 (aun (asing a2) (apow (asing a2))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25))
In Proofgold the corresponding term root is ec788c... and proposition id is d83c2d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUVctU8mrkiN8MjF1aduZG63pHg28tocvu)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ (a1 = a2)) → (¬ asubq a1 a0) → (¬ asubq a1 (asing a2)) → (¬ asubq a2 (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ asubq a3 (asing a2)) → (¬ (a3 = apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a1 (asm a4 (asing a2)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain a0 (asm a4 a2)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (apow a2) (asing (apow a2))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1)))
In Proofgold the corresponding term root is 065a80... and proposition id is 24052b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMaLSE7uGcKj9ehfFrNgpyqqwhJXQaukoQT)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a1 a0) → (¬ ain a2 a1) → asubq a1 a2 → (¬ (a2 = a3)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a2 (asing a2) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ asubq (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (apow a2)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain (apow a2) a4) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq a1 (asm a2 (asing a1)) → (asm a2 (asing a1) = a1) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq a3 (asm a4 (asing a3)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (asm a3 (asing a2) = a2)
In Proofgold the corresponding term root is 5fb7bc... and proposition id is 5dc30b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWkgrLLo1oUc1N1dcpztB1dyeTAXsH6H1T)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → ain (asing a1) (asm (apow a2) a2) → (¬ (a0 = asing a2)) → ain (asing a1) (apow (asing a1)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = a3)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (apow (asing a2)) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a1) (asm a4 a2)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain a0 (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun a4 (asing (apow a2))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 a3) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 a4) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal2 (apow (asing (asing a1))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex4 (apow a2) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a2)) a4) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aal3 (asm a4 (asing a1))
In Proofgold the corresponding term root is 4572b6... and proposition id is eb81bb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWRepEBAxWB2rn7cpvDweMXxFJ3r4GuEpL)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → ain a0 (apow a2) → (¬ asubq a1 (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (asing a1 = asm a3 (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ ain (asing a2) a3) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun a1 (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → aal2 a2 → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (asm (asm a3 (asing a1)) (asing a2) = a1)
In Proofgold the corresponding term root is 3e7bfe... and proposition id is 25f0ee...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJuQaNa2RYLDVwebxAhtUrXwph9Md2cnxH)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (¬ ain a0 a0) → (¬ ain a3 a3) → asubq a1 a2 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a0 (asm a3 (asing a1)) → asubq (asing a1) a2 → ain (asing a1) (apow a2) → (¬ (a0 = asing a1)) → (¬ asubq a1 (asing a2)) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ asubq a2 (asing a2)) → (¬ ain (asing a1) (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asm a3 (asing a1)) a2) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (asm a4 (asing a2)) → (¬ ain (apow (asing a1)) a4) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain a1 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 (asm a4 (asing a2)) → aal4 (aun a3 (asing (apow (asing a1)))) → aal4 (aun a3 (asing (apow a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → ain a3 (asm a4 (apow (asing a2)))
In Proofgold the corresponding term root is 16f1cd... and proposition id is 2b4bb0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFq7GfwoGQHz1g6Wfr8e5qE2uGpfPHAoQZ)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (¬ ain a2 a0) → (¬ ain a1 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a0 (asm (apow a2) (asing a2)) → ain a2 (asing a2) → ain (asing a1) (asm (apow a2) a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (asing a1 = a2)) → (¬ (a1 = asm a3 (asing a1))) → (¬ (asing a1 = asing a2)) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a1) a3) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ ain (apow a2) a3) → (¬ (a3 = asm (apow a2) a2)) → (¬ (asm (apow a2) a2 = apow a2)) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain a0 (asm a4 (asing a1)) → ain a0 (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a3 (asm a4 a2) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain a1 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm a3 (asing a2) = a2) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq a3 (aun (apow a2) (asing (asing a2))) → (¬ aal2 a1) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → aal3 (asm (apow a2) (asing a2)) → aal3 a3 → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24)))
In Proofgold the corresponding term root is 81340d... and proposition id is 2d02f9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdvyitcBCcqUK2vcaVrS4WxwPmZJGSQRg2)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → ain a0 a3 → ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ ain a2 a1) → (¬ (a0 = asing (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → ain a1 (apow (apow (asing a1))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a1 (apow (asm a3 (asing a1))) → ain a3 (aun a1 (asing a3)) → ain a2 (asm a4 a2) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a0 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm a3 (asing a2) = a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aex2 (apow (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 a4 → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24)
In Proofgold the corresponding term root is db0887... and proposition id is 70d7d5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSHtfhVxpF35nb8aGSpfEYwGXoHzZjraG6)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a0 (asm (apow a2) (asing a2)) → (¬ asubq a1 (asing a1)) → (¬ ain a1 (apow (asing a2))) → ain a1 (asm (apow a2) (asing a2)) → ain (asing a1) (apow (asing a1)) → (¬ ain a3 (asing a1)) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (asing a1 = a3)) → (¬ (asing a1 = apow a2)) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ asubq a3 (asing a2)) → (¬ ain (apow (asing a1)) a3) → (¬ ain (apow a2) a3) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a3 (asing a3) → (¬ ain (apow a2) a4) → ain a2 (asm a4 a2) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a1 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm a3 (asing a2) = a2) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (apow a2) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → asubq a4 (aun a4 (asing (apow a2)))
In Proofgold the corresponding term root is 4d0b75... and proposition id is 0c89cd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMCxKrsJD4Y6w5ddjy8KcrfPPSJ4GCuG6H)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a2 a1) → asubq a3 a4 → ain a2 (asing a2) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (asm a3 (asing a1))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a0 (aun a1 (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (asm a4 (asing a1)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2)))
In Proofgold the corresponding term root is 5aeb02... and proposition id is 1a1147...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFy5nGgzKTcSCHg5Tgwqx4XuZAdmmkcBVi)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a1 a4 → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → asubq a1 (asing a0) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ ain (asing a1) a0) → (¬ ain a1 (apow (asing a2))) → (¬ ain a1 (asing a2)) → asubq a1 (asm a3 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (asing (asing a1) = a3)) → (¬ (a3 = asm (apow a2) a2)) → ain a0 (apow (asing (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain a0 (asm a4 (asing a1)) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asing a2) (asing (apow a2))) → (¬ ain a3 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a2 (apow (asm a3 (asing a1))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (asm a4 a2)) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → aal2 (apow (asing (asing a1))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 a4 → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))
In Proofgold the corresponding term root is bd48b7... and proposition id is b97d64...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMWHMpfKqBWxtgUzssGwotRErQaNv4Q2Td)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → ain a2 a3 → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a1 a0) → (¬ asubq a2 a1) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a0 (apow (asing a1)) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a0 (asm a3 (asing a1)) → (¬ ain a1 (apow (asing a1))) → (¬ (asing a1 = a2)) → ain a2 (apow a2) → (¬ ain (asing a1) a3) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (apow (asing a1) = a3)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain a0 (apow (asing (asing a1))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a3 (aun (asing a1) (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a0 (asing (apow a2))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (¬ aal2 (asing a3)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal2 (asm a4 a2) → aal5 (aun (apow a2) (asing (apow a2))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1)))
In Proofgold the corresponding term root is da9d03... and proposition id is aa0a0f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMH1hZquFNd71qGgm8NXg4qJ1VdVbjMW2VF)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a1 a0) → ain a1 (asing a1) → (¬ ain (asing a2) a1) → (¬ (a0 = asm (apow a2) a2)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (asing a2) a2) → (¬ (asing a1 = apow a2)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (apow a2) a3) → (¬ (apow (asing a1) = a3)) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) a4) → ain a0 (aun a1 (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → ain (apow a2) (asing (apow a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a1) = a1) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun a1 (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ asubq a2 a0)
In Proofgold the corresponding term root is ddaeca... and proposition id is a92adc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSYuX98uuCmY7W3AYnTu9x55BuibukyqcB)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (¬ ain a1 a1) → (¬ (a1 = a2)) → (¬ (a1 = asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain (apow a2) (apow a2)) → (¬ ain (apow a2) (asing a2)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain a2 (asing a3)) → ain a3 (aun a1 (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a4 (aun a4 (asing (apow a2))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aun a2 (asing (apow a2))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2)))
In Proofgold the corresponding term root is 09b2d7... and proposition id is 89ff01...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUfAb7GaQEYHwkD3bFJ5C5AsYk3PxeUFiK)
(∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → asubq a3 a4 → ain (asing a1) (asing (asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ (a0 = asm a3 (asing a1))) → ain (asing a1) (apow (asing a1)) → ain a2 (asm a3 (asing a1)) → (¬ asubq a2 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a2))) → ain a0 (apow (asing a2)) → (¬ (a1 = asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asing a2)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing a2 = a3)) → asubq a3 (apow a2) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (asm a4 (asing a1)) → (¬ ain (asing a1) a4) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain a0 (apow (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a3 (aun a1 (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a2) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 a2) (asing a2) = asing a3) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → aal3 (asm a4 (asing a1)) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq a2 (asm a3 (asing a2))
In Proofgold the corresponding term root is 8518dd... and proposition id is 3f6eeb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYHhGbP9onEcMKvdqbeP3JXGKuBrcAnarH)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (¬ asubq a2 a1) → (¬ (a0 = a2)) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → asubq (asing a1) a2 → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a2 (asm (apow a2) a2) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (apow a2) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (asing a2 = apow a2)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun a4 (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (¬ aal2 (asing a2)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (asing (asing a2)) a4) → aex4 (aun a3 (asing (apow a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asing (asing a2)) a4) → aex5 (aun (apow a2) (asing (apow a2))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ (asing a1 = asm (apow a2) a2))
In Proofgold the corresponding term root is 262291... and proposition id is a24643...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMN5xAsdAzXowb7ggMTtuVGAb16qiqizwwg)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a0 a1 → ain a0 a3 → ain a2 a3 → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a2 a0) → ain a2 (asing a2) → (¬ ain a1 (apow (asing a1))) → asubq (asing a1) a2 → ain a2 (asm a3 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain a2 (apow (asing a2))) → (¬ ain (asing a1) a3) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (apow a2) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (asm (apow a2) a2) a3) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a2) a4) → (¬ ain a1 (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing a2))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ (asing a2 = apow a2))
In Proofgold the corresponding term root is 058fb3... and proposition id is 3eb275...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYoeFHFpmw9Jd44nfVoezYPjncQpCcdaDk)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a1 a3 → ain a2 a4 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → asubq a3 a4 → (¬ (a1 = a2)) → ain a1 (aun (asing a1) (asing (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ (asing a1 = a3)) → (¬ ain (apow a2) (apow a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (a1 = asm (apow a2) a2)) → asubq a3 (apow a2) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a0 (asm a4 (asing a2)) → (¬ ain (apow a2) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a1 (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal3 a2) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (asm a4 (asing a1)) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm a4 (asing a2)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ (a0 = a3))
In Proofgold the corresponding term root is a921af... and proposition id is 912640...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHLUvxkrRPJfPKmcLPKvxbNtX3mhA54gAq)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → ain a0 a1 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ ain a0 (asing a1)) → (¬ ain (asing a1) a1) → ain a2 (apow a2) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain a1 (asing (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asing a2 = apow a2)) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain a1 (asm a4 (asing a2)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a1 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal4 (aun a3 (asing (apow a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm a4 (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex3 (asm a4 (asing a0)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, asubq X24 a0 → (X24 = a0))
In Proofgold the corresponding term root is ef50fa... and proposition id is ba77bb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcnVAAJnmqHtWmg6J2XChAaLGpcNivJobw)
ain a0 a1 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → ain a1 a4 → ain a2 a4 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ ain a0 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (a0 = asm a3 (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ (a1 = asm a3 (asing a1))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (asing a1 = asm (apow a2) a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → ain a1 (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a2 (asing a3)) → (¬ ain (asing a2) (asing a3)) → (¬ ain a0 (asm a4 a2)) → ain a2 (asm a4 (apow (asing a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a0 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a2 (asing a1) = a1) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (apow (asing a1)) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24))
In Proofgold the corresponding term root is 05d438... and proposition id is 590cc8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGGDVmMeHGYCYn6fBTJJ9TRAWZpFn6EcGB)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (¬ ain a1 a0) → (¬ ain a2 a0) → (¬ (a0 = a1)) → (¬ ain (asing a1) a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ (a0 = asm a3 (asing a1))) → ain a2 (asm a3 (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = apow (asing a1))) → (¬ ain (apow a2) a2) → (¬ ain a2 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain (asing a2) (asing a3)) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a1 (asm a2 (asing a1)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (apow a2) (asing (asing a1))) a3 → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 a3) → aal2 (apow (asing (asing a1))) → aal2 (asm a4 a2) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 a4 → aex2 (aun a1 (asing a3)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is e7c6c7... and proposition id is 893709...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZVBatsjWsHimXVBgekBBTCwUDwN8E2SrA)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a1 a3 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ (a1 = a2)) → (¬ (a0 = a3)) → ain a0 (asm a3 (asing a1)) → (¬ asubq a1 (asing a2)) → asubq a1 (apow (asing a1)) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm (apow a2) a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → asubq a3 (apow a2) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm a2 (asing a1)) a1 → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal3 a3 → aal3 (aun (asing a2) (apow (asing a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun a3 (asing (asing a2))) → aex2 (asm a4 a2) → aex3 (asm (apow a2) (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aun a3 (asing (apow a2))) → aex5 (aun a4 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq a4 (aun a4 (asing (asm (apow a2) a2)))
In Proofgold the corresponding term root is d141ed... and proposition id is a0cdb0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMU6YvX8EtPQt6ssukMP3A6ZijtPBrb9vkW)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → asubq a1 a2 → (¬ (a0 = a1)) → ain a0 (apow a2) → (¬ ain (asing a1) a2) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain a3 (asing a1)) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (apow (asing a1)) a3) → (¬ (asing a2 = apow a2)) → (¬ (asm a3 (asing a1) = apow a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm a4 (asing a2)) → (asm a2 (asing a1) = a1) → asubq (asm a3 (asing a2)) a2 → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (asm a3 (asing a1)) → aal2 (apow (asing (asing a1))) → aal3 (asm (apow a2) (asing a2)) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1)))
In Proofgold the corresponding term root is 33de94... and proposition id is 1381e1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXTEHwAP8KZiBwGLecFDeV3aKc8WNzLkqZ)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ asubq (asing a1) (asing a2)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ (asing a1 = asing a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain a3 (asing a2)) → (¬ (a2 = asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (a3 = asm (apow a2) a2)) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → ain a0 (aun a1 (asing a3)) → ain a3 (asm a4 (asing a2)) → ain a2 (asm a4 (apow (asing a2))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm a4 (asing a3) = a3) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal2 (asm (apow a2) a2) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm a3 (asing a1)) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ (asing a1 = a2))
In Proofgold the corresponding term root is 850f0b... and proposition id is 627a53...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPPoSYM8BnG7hzoMNsojXcRsdQrFRrBz7w)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ain a0 (apow X24)) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a0 a0) → (¬ (a0 = a1)) → (¬ (a0 = a2)) → ain a1 (apow a2) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (asing a1 = asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ ain (apow a2) a3) → (¬ (asing a2 = asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain (asing a2) (asing a3)) → ain a3 (aun a1 (asing a3)) → (¬ ain (asing a2) (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (asm a2 (asing a0) = asing a1) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 a2) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal2 a2 → aal4 (aun a3 (asing (apow a2))) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (aun (asing (asing a2)) a4) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))))
In Proofgold the corresponding term root is de774a... and proposition id is d05181...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdc4n3yVt5jkSmTPUmidHS8XfvECHqL9Fb)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ (a0 = a1)) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (asing a1) → ain a0 (apow (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asing a2)) → (¬ ain (asing a2) a2) → (¬ (asing a1 = asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a1 (asm a4 (asing a2)) → adisjoint (apow (asing a1)) (asm a4 a2) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a1 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun a3 (asing (asing a2))) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 a1) → (¬ aal2 (asing a2)) → (¬ aal3 a2) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal5 (aun (asing (asing a2)) a4) → aex2 (asm a3 (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (apow a2) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (asing (asing a1))) a4) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ ain a3 (asing (apow a2)))
In Proofgold the corresponding term root is f93f15... and proposition id is 79b634...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJYNGGCHrNjRT5VdssnqH3VB8NGoNvDCpw)
ain a1 a3 → ain a0 a4 → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → ain (asing a1) (apow (asing a1)) → (¬ ain a1 (asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq (asm a3 (asing a1)) a2) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ (asing (asing a1) = a3)) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a0 (asing a3)) → ain a3 (asm a4 (asing a1)) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) a4 → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 a1) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → aal3 (asm a4 (asing a2)) → aal5 (aun (asing (asing a2)) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun a1 (asing a3)) → aex3 (asm a4 (asing a2)) → aex3 (asm a4 (asing a1)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)))
In Proofgold the corresponding term root is f4c7fa... and proposition id is 263986...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMEqk34QbcULqi1NGQPG7uRnpN4XAaTHoHv)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a0 a0) → (¬ asubq a4 a3) → (¬ (a0 = a1)) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (¬ ain a0 (asing a2)) → (¬ ain a0 (asing a1)) → ain (asing a1) (apow a2) → (¬ ain (asing a1) (asing a2)) → (¬ ain (apow a2) a2) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asing a2) a3) → (¬ (asing a2 = apow a2)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain a3 (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain (asing a2) (aun a1 (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a0 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (aun a2 (asing (apow a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25))
In Proofgold the corresponding term root is b48f20... and proposition id is 4a8d74...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVvsHcDnyT6xXm8j9mfKJNEc3QUM7ZN853)
(∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → asubq a3 a4 → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asing a2)) → ain a2 (asm (apow a2) (apow (asing a2))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asing a1) (asm a3 (asing a1))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (a3 = asm (apow a2) a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a0 (aun a1 (asing a3)) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm a2 (asing a1)) a1 → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq a4 (aun (asing (asing (asing a1))) a4)
In Proofgold the corresponding term root is 224fa7... and proposition id is af2600...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUoK1U79Wk2yw16EqaXLVE4TZdEZK2RYV3)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a0 a3 → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (¬ ain a2 a0) → (¬ ain a3 a2) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a1 (asing a1) → (¬ asubq a2 (asing a2)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm (apow a2) a2) a4) → ain a2 (asm a4 a2) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm a4 a2)) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (aun a2 (asing (apow a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))))
In Proofgold the corresponding term root is bd0f3d... and proposition id is 63d17d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMQqQ7R7L7tZXp25J46JG25inwjzWQYSbM)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (¬ ain a0 a0) → (¬ asubq a4 a3) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ ain a1 (apow (asing a1))) → ain a1 (asm (apow a2) (apow (asing a1))) → ain a2 (asm a3 (asing a1)) → (¬ ain a2 (apow (asing a2))) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a2) (apow (asing a2)) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 a3) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal3 (asm a4 (asing a1)) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1))))
In Proofgold the corresponding term root is 900cb9... and proposition id is a5fd2c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVoqWDLkQgLfQiMMPgXq24EmYVg6q3tZYm)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (apow a2) a2) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) a3) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (asm (apow a2) a2) a3) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ ain a0 (asing a2))
In Proofgold the corresponding term root is 697273... and proposition id is 6895c4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVkrtooqLgBjcD6F6VR2aGvvTZ7Hqbp9sz)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → ain a0 (apow a2) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) (asing a2)) → ain a2 (asm (apow a2) a2) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ (asing a1 = a3)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a0 = apow a2)) → asubq (asm a3 (asing a1)) (apow a2) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain a3 (apow (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a3 (aun (asing a1) (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → (asm a3 (asing a2) = a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal3 (aun a2 (asing (apow a2))) → aal5 (aun (apow a2) (asing (apow a2))) → aal5 (aun a4 (asing (apow a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (apow (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex4 a4 → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asing (asing a2)) a4) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1))
In Proofgold the corresponding term root is f723e6... and proposition id is e3a631...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMP6E9tFFSSc9gR4Sfc9cLYxz2yzVRneHR7)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → ain a0 a2 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a2 a2) → asubq a1 a2 → ain a0 (apow (asing a1)) → (¬ ain a0 (asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ ain a0 (asm (apow a2) a2)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain a3 (asing a2)) → (¬ ain (apow a2) a3) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a0 (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm a4 (apow (asing a2))) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun a2 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asing a1) (asm a2 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 a3) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (asm (apow a2) (asing a1)) → aex2 a2 → aex2 (asm a4 a2) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0))
In Proofgold the corresponding term root is 22fafd... and proposition id is 8b32df...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQNWCiDCJCJFsFKsEh7Dh9mATmMhxLmQeZ)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a2 a1) → (¬ asubq a3 a2) → (¬ ain (asing a1) (asing a2)) → (¬ ain a2 (apow (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain a2 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a2 = apow a2)) → (¬ ain (asm (apow a2) a2) a3) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a1 (asm a4 (asing a2)) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq a2 (asm a3 (asing a2)) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 a3 → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2))
In Proofgold the corresponding term root is add22d... and proposition id is 2eb568...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMY7sJFjCpf7fgK2Kt95b9shSWTAh99LC12)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a0 a2 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a2 a1) → ain a2 (asm a3 (asing a1)) → asubq a1 (asm a3 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (apow a2) a2) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain a1 (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (asm a2 (asing a0) = asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal2 (asm a3 (asing a1)) → aal2 (asm a4 a2) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (asm (apow a2) a2) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (apow (asing a1))) a4) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is 9db119... and proposition id is b744df...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKbzXs6WBWWoBEnetjNGBzqtPEHiAcvmdX)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a1 a2 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a2 a2) → (¬ (a0 = a3)) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ ain (asing a1) a0) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain a2 (asing a1)) → (¬ ain a2 (apow (asing a1))) → (¬ ain a3 (asing a1)) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a2 (asm a4 a2) → (¬ ain (asing a2) (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (asm a3 (asing a1)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → ain a2 (asm (apow a2) (apow (asing a2)))
In Proofgold the corresponding term root is e8ad18... and proposition id is 9ee203...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWznoekER34pTdTwuDdhKLrgDzJSUn7rWn)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a0 a3 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a2 a2) → (¬ (a0 = a3)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain (asing a1) (asing (asing a1)) → ain (asing a1) (apow (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ ain a2 (apow (asing a2))) → (¬ (a1 = asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a2) a4) → ain a0 (apow (asm a3 (asing a1))) → ain a0 (asm a4 (asing a2)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain a0 (asing (apow a2))) → (¬ ain (asing a2) (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (apow (asing (asing a1))) → aal4 a4 → aex3 (asm (apow a2) (asing a1)) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2)
In Proofgold the corresponding term root is 3a4360... and proposition id is 62247a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYtE6pJMqBQio9nwshfEWuEzRx5DhHAW2U)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a0 a3 → ain a1 a3 → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ asubq a4 a3) → (¬ (a0 = a1)) → asubq (asing a1) a2 → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = a3)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain (asing (asing a1)) a4) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aex2 (asm a3 (asing a2)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is cff753... and proposition id is 3450c4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJAZGrJLjgBotk4X6EV7BbJxrVS7vRUjrH)
(∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (¬ asubq a3 a2) → asubq a3 a4 → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ ain a0 (asing a1)) → (¬ ain a1 (apow (asing a1))) → ain a1 (asm (apow a2) (asing a2)) → asubq a1 (asm a3 (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ (a2 = asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing (asing a1)) a3) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (asm (apow a2) a2 = apow a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a2 (aun a3 (asing (asing a2))) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → asubq a2 (asm a3 (asing a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (asm (apow a2) (asing (asing a1)) = a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal2 (asm (apow a2) a2) → aal4 a4 → aal5 (aun a4 (asing (apow a2))) → aex2 (apow (asing a1)) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (asm a4 (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (apow a2) (asing (asing a1))) → aex3 (asm a4 (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1))))
In Proofgold the corresponding term root is 35da31... and proposition id is cd87ad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMduX9uP8ThxfnF8WmkUbJDfZ2J3fVc24jU)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a0 a0) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (¬ ain (asing a1) a0) → (¬ asubq a2 (asing a1)) → (¬ ain a1 (asing (asing a1))) → (¬ ain (apow a2) a2) → (¬ (a2 = apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (asing (asing a1) = a3)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ (asm (apow a2) a2 = apow a2)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun a4 (asing (apow a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → (asm a2 (asing a1) = a1) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ asubq (apow a2) (asing (asing a1)))
In Proofgold the corresponding term root is 3dfc4e... and proposition id is 68d3aa...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMN63N9t6eihgUvvCxQYzAmABkjHpKAy3sZ)
ain a2 a3 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (¬ ain a2 a1) → (¬ (a0 = a1)) → ain a0 (asm a3 (asing a1)) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asm a3 (asing a1))) → (¬ asubq a2 (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain (asm a3 (asing a1)) (apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ (asm a3 (asing a1) = a3)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ (asing a2 = apow a2)) → ain a0 (apow (asing (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing (asing a1)) a4) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a2 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a2 (asing a1)) a1 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal5 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm a4 (asing a0)) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → asubq a3 a4
In Proofgold the corresponding term root is d9e5c2... and proposition id is 40d565...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMEvZj36cBfJ5nedBh2bGhho7YY6u6UJN4m)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (¬ ain a0 a0) → (¬ ain a3 a2) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain a3 (asing a3) → ain a1 (asm a4 (asing a2)) → ain a3 (asm a4 a2) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a1 (aun a3 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing (apow a2))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal2 (asm (apow a2) a2) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (aun a3 (asing (asing a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 a2 → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25)))
In Proofgold the corresponding term root is f4b992... and proposition id is da564e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMX372QVrGnDYc5RjsCV9CLyUfNMbpoH3ZM)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a0 a1 → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a2 a2) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (¬ ain a1 (apow (asing a1))) → (¬ ain (asing a1) a2) → asubq a1 (apow (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) a2) → (¬ (asing a1 = asing a2)) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ (asing a2 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain a1 (apow (apow (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a2 (asm a4 (asing a1)) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (asing a2)) → (¬ ain (asm (apow a2) a2) a4) → ain a2 (asm a4 a2) → ain a3 (asm a4 (asing a1)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1)))
In Proofgold the corresponding term root is e142cf... and proposition id is 0c9a00...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKP86HcRz4wtfehatsAefqKKZ8ow8aN51i)
ain a0 a2 → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a3 a3) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ (a0 = asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = apow a2)) → (¬ (asing (asing a1) = a3)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → ain a3 (asm a4 (apow (asing a2))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (apow a2) (asing (apow a2)) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun a4 (asing (apow a2))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 a4 → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (apow (asing a1))) a4) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24))))
In Proofgold the corresponding term root is c9b8c1... and proposition id is ac6b60...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMT88Sjr4UVprvesCwi1ZKgMSYRDj5QxBH)
(∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a0 a0) → asubq a1 a2 → (¬ asubq a2 a1) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain (asing a2) a1) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (apow a2) a2) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = a3)) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ ain (apow a2) (asing a2)) → (¬ ain (apow (asing a1)) a3) → (¬ (a1 = asm (apow a2) a2)) → (¬ (a0 = apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a0 (asm a4 (asing a2)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 a2) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 (asm a3 (asing a1)) → aal5 (aun (asing (apow (asing a1))) a4) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm a3 (asing a1)) → aex3 (asm (apow a2) (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → aex2 (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is b04b78... and proposition id is 02e3eb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHpcSMBxiVbBJVJLDarNoM3vpB9PwMfWcV)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a2 (asing a2) → (¬ ain a0 (asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → asubq (asing a1) a2 → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (asing a2)) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal2 a2 → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a3)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (apow a2) (apow a2))
In Proofgold the corresponding term root is a3ed10... and proposition id is c39e96...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMaKNp46JVNz1TWmcexfEStVSHrEEuRmV1N)
(∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a2 a4 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a2 a0) → ain a1 (asing a1) → (¬ ain (asing a1) a2) → ain a1 (asm (apow a2) (apow (asing a2))) → ain a1 (asm (apow a2) (asing a2)) → asubq a1 (apow (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ (a1 = asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ asubq a3 (asing a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a2) (asing (asing a2)) → ain a0 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a1 (asm a4 (asing a2)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a0 (aun a3 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing a1)) a4) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex4 (aun a3 (asing (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1)))))
In Proofgold the corresponding term root is f5f0d5... and proposition id is 7762ca...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWWVkSmYFRSPXrWomRALMusxZbDwBGtAXj)
ain a0 a2 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asing a2)) → (¬ ain a0 (asing (asing a1))) → (¬ ain a0 (asm (apow a2) a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → ain a0 (aun a1 (asing a3)) → (¬ ain a0 (asm a4 a2)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 a3) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → aal4 (aun a3 (asing (apow (asing a1)))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (asm a3 (asing a1)) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm a4 a2) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (apow a2) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex5 (aun (asing (asing a2)) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3))
In Proofgold the corresponding term root is 292998... and proposition id is 8a6c8b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYBr98breLJ1m587NCFSeprGeyd5VnJxyG)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a1 a0) → asubq a2 a3 → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → asubq (asing a1) a2 → ain (asing a1) (asm (apow a2) a2) → ain a1 (asm (apow a2) (asing a2)) → asubq a1 (apow (asing a1)) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ (a2 = apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (asing a2 = a3)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (apow a2) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asing a2) (asing (apow a2))) → (¬ ain a3 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (asm a4 a2) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asing (asing a1)) a4) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (apow a2) (asing (apow a2))) → aex3 (aun a2 (asing (apow a2))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is 0465ea... and proposition id is 109934...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVZGh9TEvusKfbEoswhhNxuV2cZJpcKwu8)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → ain a0 a4 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ ain a0 a0) → (¬ ain a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ ain (asing a1) a0) → (¬ (asing a1 = a2)) → asubq a1 (asm a3 (asing a1)) → (¬ ain a2 (apow (asing a2))) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (asm (apow a2) a2 = apow a2)) → (¬ ain a1 (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → ain a3 (asm a4 (asing a2)) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a0 (asm a4 a2)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a1 (asm a2 (asing a1)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal5 (aun a4 (asing (apow a2))) → aex2 (asm a3 (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2))
In Proofgold the corresponding term root is aea781... and proposition id is 0d5116...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKJg9M4PtabRCZpkss8R9gUvFFWeLLo4uh)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → ain a1 a4 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (¬ ain a2 a1) → (¬ ain a3 a2) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (apow (asing a1))) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) (asing a2)) → ain a2 (asm (apow a2) a2) → (¬ ain a0 (asm (apow a2) a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (asing a1 = apow a2)) → (¬ ain (asing a2) (apow a2)) → (¬ asubq a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a0 (apow (asing (asing a1))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (asing a2) a4) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain (apow a2) a4) → adisjoint (apow (asing a1)) (asm a4 a2) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm a4 (asing a3)) a3 → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal2 a2 → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm a4 a2) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (apow a2) → aal4 (aun a3 (asing (apow (asing a1)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2))
In Proofgold the corresponding term root is 24b07d... and proposition id is 0cb541...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVC5xPRGZQByUVGTzYbGnex6oZJFbGZ7k1)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → asubq a3 a4 → ain (asing a1) (apow (asing a1)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain (asing (asing a1)) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a1 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a3)) → (¬ aal3 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (asm (apow a2) (asing a1)) → aal3 (aun (asing a2) (apow (asing a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1)
In Proofgold the corresponding term root is 337c08... and proposition id is 667cbe...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWNys2gNN7kEqUmNQQzY9cuE4dq2CT6fVo)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a1 a3 → ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → asubq a2 a3 → (¬ (a1 = a3)) → (¬ ain (asing a1) (asing a1)) → (¬ ain (apow a2) a2) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (a2 = asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2))
In Proofgold the corresponding term root is 3783cc... and proposition id is e54705...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXGCbk3nEAFRERSBVekXPgzAfpu81EwyCi)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (¬ (a0 = a1)) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a0 (asm a3 (asing a1)) → (¬ (a0 = apow (asing a1))) → (¬ ain (asing a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing a2) a3) → (¬ ain a3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow a2) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a1 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a1 (asm a2 (asing a1)) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (asm a4 (asing a3) = a3) → asubq a3 (aun (apow a2) (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 a1) → (¬ aal2 (asing a3)) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aex2 (asm a3 (asing a1)) → aex2 (aun a1 (asing (apow a2))) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm (apow a2) (asing a0)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))))
In Proofgold the corresponding term root is 590e7f... and proposition id is 2cc393...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHrViutVJ7r4rDjmHAB1vMCSLCkXWLLxMU)
(∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → asubq a1 (asing a0) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (¬ ain (asing a1) a2) → ain (asing a1) (apow (asing a1)) → asubq a1 (apow (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a1))) → (¬ ain (asing a1) (asing a1)) → (¬ ain a1 (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ ain (asing a2) a3) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (asm a3 (asing a1) = a3)) → (¬ (a3 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asing a2 = apow a2)) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain a0 (asm a4 (asing a1)) → ain a2 (asm a4 (asing a1)) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun (apow a2) (asing (apow a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24))))
In Proofgold the corresponding term root is 318dad... and proposition id is 81012b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPb6rCicfcSi7AbfstCjmjjj7WyNSU7nkW)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (asing a1)) → (¬ (a1 = asing a2)) → (¬ asubq (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ ain (apow (asing a1)) a3) → asubq (asm a3 (asing a1)) (apow a2) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (a3 = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain a3 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq a1 (asm a2 (asing a1)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (¬ aal2 (asing a1)) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal3 (asm (apow a2) (asing a1)) → aal3 (aun a2 (asing (apow a2))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal5 (aun (asing (asing a1)) a4) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun a1 (asing (apow a2))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is fd808d... and proposition id is 874f00...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMY9kEivv9B3BDBGpAmbRZVhMPtNpig7wZ3)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ asubq a4 a3) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (asm (apow a2) (asing a2)) → (¬ asubq a2 (asing a1)) → (¬ asubq a2 (asing a2)) → (¬ (a1 = asm a3 (asing a1))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing (asing a1)) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ (a3 = apow a2)) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain (asing a2) (asing a3)) → (¬ ain (apow (asing a1)) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a3 (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a2 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a2)) a2 → asubq a2 (asm a3 (asing a2)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 a3) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun a3 (asing (apow a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (apow (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1)))
In Proofgold the corresponding term root is 2259a9... and proposition id is 649bf0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTrct2VVdsCS236Gz2uL2umHS3vvHDnMdB)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a1 = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a3 (asing a3) → (¬ ain (asm (apow a2) a2) a4) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal3 (apow (asing a1))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → aal3 (aun (asing a1) (apow (asing a2))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (asm a4 (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25))
In Proofgold the corresponding term root is 75b9dd... and proposition id is fe5a2f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMT8NSyYrv6kpSehNJrsiiiLBCaYHZ45W9n)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ (a0 = a3)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain (asing a1) (asm (apow a2) a2) → (¬ ain a1 (apow (asing a2))) → (¬ ain (asing a1) a1) → (¬ ain (asing a2) a1) → (¬ ain a2 (asing a1)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain a3 (apow (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a0 (asm a4 (asing a2)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq a1 (asm a2 (asing a1)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 a3 → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (aun (asing a1) (asing (asing a1))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (apow (asing a1))) a4) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1)))
In Proofgold the corresponding term root is caa824... and proposition id is 74bd37...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXadqZx69mbyhti26pYFK9eiBkc4MG7S1b)
(∀X24 : set, (¬ ain X24 X24)) → ain a0 a3 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (¬ asubq a4 a3) → (¬ asubq a2 a0) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ ain (asing a1) a0) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ (a0 = asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ (asing a1 = a2)) → (¬ (a1 = apow (asing a1))) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (apow (asing a1) = a3)) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain a0 (apow (asm a3 (asing a1))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a1 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal4 a3) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal3 a3 → aal4 (aun a3 (asing (asing a2))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex5 (aun a4 (asing (apow a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (asm (apow a2) (asing (asing a1)) = a3)
In Proofgold the corresponding term root is 7fc7f8... and proposition id is aee9cf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMctGS21Az7PG9mF2rRXEwdbgBiF8SghUJd)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (¬ ain a0 a0) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a0 (apow (asing a1)) → (¬ ain a0 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ asubq (asing a2) a2) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (apow (asing a1)) a3) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (asm a4 a2) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing a2)) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing a1) (asing a3))) → aal2 (asm a3 (asing a1)) → aal5 (aun (asing (asing a2)) a4) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2)
In Proofgold the corresponding term root is a60025... and proposition id is 1fc68a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVnbtrpt1SVm7YmTokUMptG43ivF13Geob)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (asing a1) → (¬ ain a1 (asing a2)) → asubq a1 (apow (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (asing (asing a1)) a3) → (¬ (asing a2 = apow a2)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → (¬ ain a2 (asing a3)) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a1 (aun (asing a1) (asing a3)) → ain a3 (asm a4 (asing a2)) → ain a3 (asm a4 (asing a1)) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a2)) a2 → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal3 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (apow a2) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (asing (asing (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (aun (asing a2) (apow (asing a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex2 (aun (asing a2) (asing (asing a2)))
In Proofgold the corresponding term root is e41913... and proposition id is 91d3f7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNeKifZUYs8PBAENY6w7A6xPq8Vc8yMJcu)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (¬ ain a0 a0) → (¬ ain a2 a0) → (¬ ain a0 (asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ (a0 = asing (asing a1))) → asubq a1 (apow (asing a1)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a2 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a1))) → ain a2 (asm (apow a2) (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asing a2 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a2) a4) → ain a0 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a3 (asm a4 (asing a2)) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (asm a3 (asing a2) = a2) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 a4) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow a2) (asing a1)) → aex3 (asm a4 (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24)
In Proofgold the corresponding term root is d94568... and proposition id is f42aad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUeUagu9qxCgj5k4KkigLJ9nsj5xGTA1Fd)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a0 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ asubq a1 (asing a2)) → (¬ (a1 = asing (asing a1))) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = apow a2)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (apow (asing a1)) a4) → ain (asing a2) (aun (asing (asing a2)) a4) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing a2)) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 a2 → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 a4 → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (aun (asing (asing a2)) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is fb8a70... and proposition id is 5d5c5f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKQ49NdiUTTU3hqgewukr2sDQSaACtNaHf)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → ain a0 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a2 a2) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (¬ asubq a1 (asing a2)) → (¬ ain a2 (asing a1)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain a0 (apow (asing (asing a1))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain (asing a2) (aun a1 (asing a3))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → aal4 (apow a2) → aal4 (aun a3 (asing (asing a2))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 a4 → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain (asing a2) (aun (apow a2) (asing (asing a2)))
In Proofgold the corresponding term root is 59ad56... and proposition id is 1bcbd8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXJnnt2ECBj3odwaS6SnoeMDwAh3FkP2F8)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (asm (apow a2) (apow (asing a2))) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm a3 (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (a1 = asing (asing a1))) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asing a2 = apow a2)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a0 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aal2 (asm a4 a2) → aal3 a3 → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → aal5 (aun (apow a2) (asing (asing a2)))
In Proofgold the corresponding term root is 9f782b... and proposition id is 915c93...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSdfUwVQc95NoyMTZueQ4Q7UKRWgRdY2N8)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → ain a1 a2 → ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a2 a1) → asubq a1 a2 → (¬ (a1 = a3)) → ain a1 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (apow (asing a1)) → (¬ ain a3 (asing a1)) → (¬ ain (asing a2) a2) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain a0 (asm a4 (asing a2)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (asm a4 a2) → ain a3 (asm a4 (apow (asing a2))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun a1 (asing (apow a2))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (asm a4 (asing a0)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3)))
In Proofgold the corresponding term root is 7118ee... and proposition id is c070d8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPc9b1zx2hgyN61UmcDwFTvw4H4rwvzFoe)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ (a0 = a2)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain a1 (asing a2)) → asubq a1 (apow (asing a1)) → (¬ ain a1 (asing (asing a1))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ (asing a1 = asing a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (apow a2)) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asing a2 = asm (apow a2) a2)) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain a2 (asing a3)) → (¬ ain (asing a2) (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a2 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (asing (apow a2)) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (asm (apow a2) (asing a2)) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex4 (apow a2) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25)))
In Proofgold the corresponding term root is f51705... and proposition id is 20a757...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSUBUbheE8r8RZgtALD7KhHB34pMQVUob9)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a4 a3) → (¬ (a0 = a1)) → (¬ ain a0 (asing a2)) → (¬ ain a0 (asing a1)) → (¬ ain (asing a1) a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = asing (asing a1))) → (¬ ain (asing a2) a1) → (¬ (a1 = asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (¬ (a2 = asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (asm a4 a2) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (apow a2) (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm a4 a2) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun (asing (asing (asing a1))) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25)))
In Proofgold the corresponding term root is 013d26... and proposition id is dd167a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJKaEgc36c6JEg6BTeZfKufgewQXhf1761)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a2 a2) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a0 (apow (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ ain (apow a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → asubq a3 (apow a2) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm a2 (asing a0) = asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 (asing (apow a2))) → (¬ aal4 a3) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → aal3 (asm a4 (asing a2)) → aex2 (asm a3 (asing a1)) → aex3 (asm a4 (asing a1)) → aex4 (apow a2) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex5 (aun (asing (asing a2)) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))
In Proofgold the corresponding term root is ad7184... and proposition id is de71ce...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYvQQgwpeKEgB4jVdbAdvLd1WqzGXeXciQ)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ (a0 = a3)) → (¬ ain a1 (apow (asing a1))) → (¬ (a0 = asm a3 (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain a2 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asing a2)) → (¬ (asing a2 = a3)) → (¬ (a3 = asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a3 (aun a3 (asing (apow a2))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → aal3 (asm (apow a2) (asing a2)) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aun a1 (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex5 (aun (asing (asing a2)) a4) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is a80ddc... and proposition id is 849ef9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQzV7mGXb6EANARx8jpYKLKbQXj6NYzdZP)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a3 a4 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain a1 (apow (asing a2))) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (a0 = asing a2)) → ain a2 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ asubq (asing a2) a2) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a0 (asm a4 (asing a1)) → (¬ ain a2 (aun a1 (asing a3))) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain (asing a1) (aun (asing (asing a1)) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (apow a2) (aun a2 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → (asm a2 (asing a0) = asing a1) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aex2 (aun a1 (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex5 (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (¬ asubq (aun (asing (asing a2)) a4) a4)
In Proofgold the corresponding term root is a42dec... and proposition id is 42a2f8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUxgWDCugPkbCE6Y8eDzaFccZoHQuRuRi8)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → ain a1 a2 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a1 (aun (asing a1) (asing (asing a1))) → ain (asing a1) (asm (apow a2) a2) → (¬ ain (asing a2) a1) → (¬ ain a0 (asing (asing a1))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain a0 (asm a4 a2)) → ain a1 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a2)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) a4 → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 a4) → aal2 (apow (asing (asing a1))) → aal4 (aun a3 (asing (asing a2))) → aal5 (aun (asing (asing a1)) a4) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 a2 → aex2 (apow (asing a1)) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (apow (asing a2)) → aex2 (asm a3 (asing a2)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm a4 (asing a0)) → aex3 (asm a4 (asing a3)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is 169873... and proposition id is ed38d2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJQgnQ3xxR4RkjdpsoWxdTfkkVhvwXSgyA)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a0 a2 → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a2 a0) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a2 (apow a2) → (¬ ain a0 (asm (apow a2) a2)) → (¬ asubq a2 (asing a2)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (apow a2) (asing a2)) → (¬ asubq a3 (asing a2)) → asubq (asm a3 (asing a1)) (apow a2) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a0 (asing a3)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a2)) a4) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a4 (aun (asing (asing a2)) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (apow a2) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex5 (aun (apow a2) (asing (asing a2))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → adisjoint a2 (aun (asing (asing a1)) (asing a3))
In Proofgold the corresponding term root is 9e9550... and proposition id is 885568...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVCfzzKCBMVUHjtRyHU4vexmvHCQowCyYm)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a1 a0) → (¬ ain a2 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ asubq a1 (asing a1)) → ain (asing a1) (asm (apow a2) a2) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain (asing a1) (apow (asing a2))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (a1 = asing (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing a2)) → (¬ (a2 = asing a2)) → (¬ (asm a3 (asing a1) = a3)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asm (apow a2) a2 = apow a2)) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a2 (asing a3)) → (¬ ain (asing a2) (aun a1 (asing a3))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asing a2) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 (apow (asing a1))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm a3 (asing a2)) → aex4 (apow a2) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 7b5f16... and proposition id is 19e820...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYhHErjWYZd7QpD3GkRdGuUYULryHSZWbM)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ ain (asing a1) a0) → (¬ ain a1 (apow (asing a1))) → (¬ ain (asing a2) a1) → ain a2 (asm (apow a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a1 = apow a2)) → (¬ (apow (asing a1) = a3)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain (asing a1) (asm a4 a2)) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal3 a2) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → aal3 a3 → aal4 a4 → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aun (asing a1) (asing (asing a1))) → aex3 (asm (apow a2) (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 a4 → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ ain (asing a2) (asm (apow a2) (asing a1)))
In Proofgold the corresponding term root is 456353... and proposition id is de0a1e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRcszcJJTeA814irvYbnP4qEQHwsspiuXx)
(∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ asubq a1 a0) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (¬ ain (asing a1) a0) → (¬ ain a1 (apow (asing a1))) → ain a2 (apow a2) → (¬ ain a2 (apow (asing a1))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ (asm (apow a2) a2 = apow a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain (asing a2) (aun a1 (asing a3))) → (¬ ain a0 (asm a4 a2)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun a3 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → asubq a2 (aun a2 (asing (apow a2))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → aal5 (aun a4 (asing (apow a2))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (asm a4 (asing a3)) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2))))
In Proofgold the corresponding term root is 555fbd... and proposition id is a9b833...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPrETM75XDEPzsM5JF3ZiCRxEisX3WQoSJ)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (¬ ain a0 a0) → (¬ (a0 = a1)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a0 (asm (apow a2) (asing a2)) → asubq a1 (apow (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a1 = asing (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain (asing a2) (asing a3)) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow a2) a4) → (¬ ain (asing a1) (asm a4 a2)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) a4 → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal3 (asm a4 (asing a1)) → aal4 (aun a3 (asing (asing a2))) → aal4 a4 → aal5 (aun a4 (asing (apow a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm a4 a2) → aex2 (aun (asing a3) (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1)))
In Proofgold the corresponding term root is cc3a64... and proposition id is 00c956...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJaMmS2aBwHX4AzDnTuc5NcnFyCWHfJAvW)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (¬ ain a0 a0) → ain a1 (apow a2) → (¬ (a0 = asing a1)) → (¬ ain (asing a1) a1) → (¬ ain a2 (asing a1)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (a2 = apow a2)) → (¬ (a0 = apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → asubq a3 (apow a2) → ain a1 (apow (apow (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a1 (apow (asm a3 (asing a1))) → ain a3 (asing a3) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm a4 (asing a2)) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (asm a2 (asing a0) = asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq a2 (asm a3 (asing a2)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal3 (aun a2 (asing (apow a2))) → aal4 (apow a2) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (apow (asing a2)) → aex2 (aun a1 (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ ain a2 (apow (asing a1)))
In Proofgold the corresponding term root is 22b111... and proposition id is c4719f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMacRtx1o7uApJvpJiouatq27vTe8iwT5E4)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (¬ (a1 = asing a2)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing a2 = a3)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a0 (aun a1 (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing a2)) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a4 (aun a4 (asing (apow a2))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a3 (asing a2) = a2) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq a3 (aun (apow a2) (asing (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 a2) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (aun a2 (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1))))
In Proofgold the corresponding term root is f5b6bf... and proposition id is 3a3f63...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFvCHVgrQ974Vb6svwS6m2qcAiKYZg1Gcn)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → asubq a3 a4 → (¬ asubq a4 a3) → (¬ (a0 = a2)) → ain a0 (asm (apow a2) (asing a2)) → ain a1 (apow a2) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (asing a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain (asing a1) (asm a4 a2)) → ain a2 (asm a4 (apow (asing a2))) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (asm a3 (asing a2) = a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 a4) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 (asm (apow a2) a2) → aal4 a4 → aex2 (aun a1 (asing a3)) → aex2 (asm a4 a2) → aex3 (asm (apow a2) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun a4 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25)
In Proofgold the corresponding term root is 53c8c8... and proposition id is 274f9b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUhJdFqYo1noPyMNkeNtP8BMC2xVvzqDaz)
(∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (¬ ain (asing a1) a0) → ain a1 (asing a1) → ain a0 (apow (asing a1)) → (¬ (a1 = asing a1)) → ain (asing a1) (asm (apow a2) a2) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain (asing a2) a1) → (¬ (asing a1 = asing a2)) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ (a1 = apow a2)) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain a2 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (asing a1) (asm a4 a2)) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain a0 (aun a1 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (asm a2 (asing a0) = asing a1) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq a2 (apow (asm a3 (asing a1))) → (¬ aal2 (asing a2)) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aex2 (asm (apow a2) a2) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (asm a4 (asing a2)) → aex3 (asm a4 (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3)))
In Proofgold the corresponding term root is 56cc92... and proposition id is 9eb6ac...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLvpiwTKkfLxXUUitUdzgB7ysAdoCU8QcH)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → asubq a1 a2 → ain a0 (apow (asing a1)) → ain (asing a1) (apow a2) → (¬ (a0 = asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ (asing a1 = a2)) → (¬ ain a2 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → ain a0 (apow (asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun a3 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (apow (asing (asing a1))) → aal3 (asm a4 (asing a1)) → aex2 a2 → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ ain (apow a2) (aun a3 (asing (asing a2))))
In Proofgold the corresponding term root is 9b7422... and proposition id is 89275e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLs9xDW6zN5MEZ8EX46meQXw1QNqtytCUP)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → ain a1 a2 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a2 a2) → (¬ asubq a4 a3) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a2 (asing a2) → (¬ ain (asing a1) a2) → (¬ (a0 = asing (asing a1))) → (¬ ain a2 (apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) a3) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asing a2) a4) → ain a0 (aun a3 (asing (asing a2))) → ain a1 (aun (asing a1) (asing a3)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a1) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm a4 (asing a3) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing a2)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun a3 (asing (apow a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm a3 (asing a1)) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is bcb149... and proposition id is 5ed2ce...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJLD9yPrCfbyrZSLmRjL2gkVBqT482ynNi)
(∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ ain a2 a0) → (¬ asubq a2 a1) → (¬ ain a0 (asing a2)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain a0 (asing (asing a1))) → (¬ ain a3 (asing a1)) → (¬ (asing a1 = asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) a3) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (asing a2) a4) → ain a3 (asing a3) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal6 (aun a4 (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex2 a2 → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex4 (aun a3 (asing (asing a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1)))
In Proofgold the corresponding term root is 43fbaa... and proposition id is d1b5c1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMT5SJErDgH6dY7nTc99XNpqFCymnnZYMQN)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (¬ ain a1 a1) → (¬ ain a2 a1) → (¬ ain (asing a1) (asing a2)) → (¬ ain (asing a2) a2) → (¬ ain (asm a3 (asing a1)) a2) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a1 (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain (asing a2) a4) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a3 (aun a1 (asing a3)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (asm a2 (asing a1) = a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a3 (asing a1)) a4 → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 a4) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (asm (apow a2) a2) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 a4 → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (apow a2) (asing (apow a2))) → aex5 (aun a4 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26))
In Proofgold the corresponding term root is c8fd21... and proposition id is 203d75...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNJGBcmK8KmcpjhzqLaU5BTweKtAsS439S)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a0 a3 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → asubq (asing a0) a1 → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a1) → asubq a1 a2 → (¬ (a1 = a3)) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (apow a2) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain a3 (asing a1)) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain a1 (asing (apow (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a2 (aun a1 (asing a3))) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain (apow (asing a1)) a4) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a2) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal3 (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 (asm (apow a2) (asing a2)) → aal4 (apow a2) → aal4 (aun a3 (asing (apow (asing a1)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm a4 (asing a2)) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))))
In Proofgold the corresponding term root is c06abb... and proposition id is 4badfd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXmThPPa2NzBnqfjbgTDK9SU7ngJS5UQym)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (asing a1) → ain a0 (asm (apow a2) (asing a2)) → (¬ ain (asing a1) a2) → (¬ (a0 = asing a2)) → (¬ asubq a2 (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ ain a3 (asing a2)) → (¬ (a2 = apow a2)) → (¬ ain (asing a2) a3) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (asing a2 = a3)) → (¬ (asm a3 (asing a1) = apow a2)) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a3 (aun (asing a1) (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a0 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → aal2 a2 → aal2 (asm a3 (asing a1)) → aex2 (asm (apow a2) a2) → aex2 (aun a1 (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 a4 → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → ain a0 (aun a2 (asing (apow a2)))
In Proofgold the corresponding term root is a003ad... and proposition id is 22664d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKG2m83bgMV8RWzw86kVhvDjGMzr5mtsq1)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ (a0 = a2)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain a0 (asing a1)) → asubq (asing a1) a2 → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a2 (asing a1)) → (¬ ain a2 (apow (asing a1))) → (¬ ain a0 (asm (apow a2) a2)) → (¬ (a1 = asing a2)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = a3)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (asing a2)) → (¬ ain (asing a2) a3) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) a3) → asubq a3 (apow a2) → (¬ (a3 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (a3 = apow a2)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a2) a4) → ain a1 (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain (apow a2) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq a2 (asm a3 (asing a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal5 (apow a2)) → (¬ aal5 a4) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (asm a3 (asing a1)) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2))
In Proofgold the corresponding term root is bd9999... and proposition id is 0056b7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTr2JDXRQ57CE9BMMfxVEhoQ5S9JCpFwMk)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → ain a0 a1 → ain a1 a2 → ain a3 a4 → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (¬ (a1 = a2)) → ain a2 (asing a2) → ain a0 (asm a3 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ asubq a1 (asing a2)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain a3 (asing a2)) → (¬ (a2 = apow a2)) → (¬ asubq a3 (asing a2)) → (¬ ain (apow (asing a1)) a3) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (asing a2 = a3)) → (¬ (asing a2 = asm (apow a2) a2)) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (asing a2)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aex2 (apow (asing a1)) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (apow a2) (asing a2)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is dd0377... and proposition id is 0ac5c6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMT9ecgysxQxnPrMdYXpU9nnXRDEHYYUnxh)
ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → asubq a1 a2 → (¬ (a0 = a1)) → (¬ (a2 = a3)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain a2 (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ ain a3 (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a1 (asm a2 (asing a1)) → (asm a3 (asing a2) = a2) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aex2 (apow (asing a1)) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex5 (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))
In Proofgold the corresponding term root is da21b3... and proposition id is 913246...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcFVbtB6Q1N3Rma1RDhfXnR3vNYckyRfz2)
ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (¬ ain a1 a1) → (¬ asubq a2 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain a0 (apow a2) → ain (asing a1) (apow a2) → (¬ ain a0 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (apow a2) (asing a2)) → (¬ ain (apow (asing a1)) a3) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (apow (asing (asing a1))) → ain a0 (apow (apow (asing a1))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a1 (asing a3)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (asm a3 (asing a2) = a2) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal3 (asm (apow a2) (asing a2)) → aal4 a4 → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal5 (aun (asing (asing a1)) a4) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm a3 (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 a4 → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24)
In Proofgold the corresponding term root is 35871e... and proposition id is 39e609...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLBtQC2Rw2nwumv9MQ1CSpB2Nnn2pe5r8A)
(∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (¬ asubq a4 a3) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (¬ ain a1 (asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ asubq (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → asubq a3 (apow a2) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → ain a0 (aun a3 (asing (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain a3 (asm a4 (apow (asing a2))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal2 (apow (asing (asing a1))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (apow (asing a1))) a4) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3)
In Proofgold the corresponding term root is 7b0476... and proposition id is 88b7b5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMKeKWY4kjkPdGhomjEsn2ynJKEmY4DUNji)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain (apow a2) (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a3 (asing a3) → ain a3 (asm a4 (asing a1)) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → (¬ ain (asing a1) (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal3 a3 → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun (asing (apow (asing a1))) a4) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))))
In Proofgold the corresponding term root is db9484... and proposition id is 0ec814...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYoC23fJLhzRzfeY17XDdc9e2YRVdAPyJ5)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a1 a4 → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a3 a2) → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → asubq (asing a1) a2 → (¬ (a0 = asing a1)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain a3 (apow a2)) → (¬ ain a3 (asm a3 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a2 (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) a4 → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal2 a1) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (apow a2) (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25)))
In Proofgold the corresponding term root is 8c836a... and proposition id is f90616...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMF3d54fydsPiBwo9DQFnspW7sucroKXnYS)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a0 a2 → ain a1 a4 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ asubq a1 (asing a1)) → (¬ (a0 = asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ asubq a2 (asm a3 (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ (asm a3 (asing a1) = a3)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ (asing a2 = apow a2)) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a3 (asing a3) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm a4 a2) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain a0 (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a2)) a2 → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq a2 (apow (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex5 (aun (asing (asing a2)) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ (apow (asing a1) = a3))
In Proofgold the corresponding term root is 7ff0b8... and proposition id is 9b54bc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYXnD5fTjeLeuyagWWHjoQNX2WrzdETiZn)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → asubq a1 a2 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain (asing a1) (asing (asing a1)) → ain a0 (asm a3 (asing a1)) → (¬ ain (asing a1) (asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ asubq (asing a2) a2) → (¬ ain (apow a2) (asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (asing (asing a1)) a3) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a3 (aun a1 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (asm a3 (asing a1)) a4 → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing a2)) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 a3 → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → ain (asing a2) (aun (asing (asing a2)) a4)
In Proofgold the corresponding term root is 8d72d2... and proposition id is 68512f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSfRE9Qf3xYymSjmKeQp7afGixZD83QDHq)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ (a0 = asing a1)) → (¬ (a1 = asing a1)) → (¬ (a0 = asing a2)) → ain a0 (apow (asing a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) a2) → (¬ ain (apow a2) a2) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → asubq a3 (apow a2) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (asm a4 (asing a2)) → ain a3 (asm a4 (asing a2)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain a0 (asing (apow a2))) → (¬ ain a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 a3) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 a4) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal2 (asm a3 (asing a1)) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm a4 (asing a1)) → aex3 (aun a2 (asing (apow a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25))
In Proofgold the corresponding term root is 4b6272... and proposition id is f3d733...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZVoU4eBL7oFTiDVFm56CXgfiBKDekeigJ)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (¬ ain (asing a1) a0) → ain a0 (apow (asing a1)) → ain a1 (asm (apow a2) (asing a2)) → (¬ (asing a1 = a2)) → ain (asing a1) (asm (apow a2) (asing a1)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) a3) → (¬ asubq (apow a2) (asing a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a1 (apow (apow (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain (asing a2) a4) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a0 (aun a3 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm a3 (asing a2) = a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (apow (asing a2))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a1 a1)
In Proofgold the corresponding term root is 3db287... and proposition id is d0889a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXPzr1pssysWEJSKyPi8SwN57smgtsA64r)
ain a2 a4 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ ain a3 a2) → asubq a1 a2 → (¬ (a1 = a3)) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (¬ asubq a1 (asing a1)) → (¬ ain (asing a2) a1) → (¬ ain (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ (asing a1 = asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain a3 (apow a2)) → (¬ ain (asm (apow a2) a2) (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ ain (asing (asing a1)) a3) → (¬ (a1 = apow a2)) → (¬ (asing (asing a1) = a3)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a1 (asing a3)) → (¬ ain a2 (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a0 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing a2)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → aal3 (asm (apow a2) (asing a1)) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm a3 (asing a1)) → aex2 (apow (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun a3 (asing (apow a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1))
In Proofgold the corresponding term root is 1a3891... and proposition id is 000a1e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWsi9cqmZ8wKqK6USEKA6LtfoDE5KGhzpA)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a1 a0) → (¬ ain a2 a1) → ain (asing a1) (asm (apow a2) a2) → ain a1 (asm (apow a2) (asing a2)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain a3 (asing a1)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ (apow (asing a1) = a3)) → asubq a3 (apow a2) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain (apow a2) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a1 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (¬ aal2 (asing a2)) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → aal2 a2 → aal2 (apow (asing (asing a1))) → aal3 (asm a4 (asing a2)) → aal5 (aun (asing (asing a2)) a4) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (apow (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (apow a2) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 6e3b97... and proposition id is a48070...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXX4gQqRwN26FJkKmhAXRtsEF6CDrZXaGd)
(∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a2 (asing a2) → (¬ asubq a1 (asing a1)) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain a0 (asing (asing a1))) → ain (asing a1) (apow (asing a1)) → ain a2 (asm a3 (asing a1)) → (¬ ain a2 (apow (asing a2))) → (¬ ain (apow a2) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) a3) → ain a0 (apow (apow (asing a1))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain (apow a2) a4) → ain a1 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain (asing a2) (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → asubq a1 (asm a2 (asing a1)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing a2)) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing a1) (apow (asing a2))) → aal5 (aun (asing (apow (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (aun (asing (apow (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ ain a1 (asm a3 (asing a1)))
In Proofgold the corresponding term root is 1b531d... and proposition id is 2e95bd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTxoZk7vCcx4Zc63vwRZ5C3gCA1tCHKYY9)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ asubq a2 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ ain a0 (asing a2)) → ain a0 (asm a3 (asing a1)) → ain (asing a1) (asm (apow a2) a2) → (¬ (asing a1 = a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm a3 (asing a1)) → (¬ (a1 = asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ (a2 = apow a2)) → (¬ (a2 = asing a2)) → (¬ asubq a3 (asing a2)) → (¬ ain (asing (asing a1)) a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a2) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a0 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (asm a4 (asing a3) = a3) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal5 (apow a2)) → aal3 (asm (apow a2) (asing a2)) → aal5 (aun (asing (asing a1)) a4) → aex2 (asm (apow a2) (apow (asing a1))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24))))
In Proofgold the corresponding term root is 602bf5... and proposition id is 7e0277...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFoRMHKhZUDsvC7f2FQm6sZc68ECJZHnWW)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → ain a0 (asm (apow a2) (asing a2)) → ain a1 (apow a2) → ain a2 (asm (apow a2) a2) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (a3 = apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 a2) (asing a3) = asing a2) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal2 (asing a1)) → (¬ aal3 (apow (asing a2))) → (¬ aal5 a4) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal4 (aun a3 (asing (asing a2))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing a1)) → aex3 (asm a4 (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex5 (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a0 (asing (apow a2)))
In Proofgold the corresponding term root is 8dc0f9... and proposition id is 6fee1f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZSuzGLegHu58Dqbbkz1GgG7mKHofbfiD5)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ (a0 = a1)) → (¬ (a1 = a3)) → (¬ asubq a1 a0) → ain a0 (asm (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a2))) → (¬ (a1 = asm a3 (asing a1))) → (¬ asubq (asing a1) (asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ asubq a2 (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (a2 = asing a2)) → (¬ (asing a2 = apow a2)) → ain a1 (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain (asing a1) (aun (asing (asing a1)) a4) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a3 (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (asm a4 (asing a3)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2)))
In Proofgold the corresponding term root is f228e1... and proposition id is 3cdddd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJvCWmRevG8icjVEpBJRtFGkHCJ3d19o2h)
(∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain (asing a1) a0) → ain (asing a1) (asing (asing a1)) → asubq (asing a1) a2 → ain a1 (asm (apow a2) (apow (asing a2))) → ain (asing a1) (apow (asing a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (apow (asing a1)) a3) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a1 (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a2)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a2 (asm a4 a2) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq a3 (asm a4 (asing a3)) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25)))
In Proofgold the corresponding term root is 407dd1... and proposition id is dee63f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPzYTMjxdo9nRwScxEHbEotrJD8JF8ZbKx)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (¬ ain a1 a1) → (¬ asubq a2 a1) → (¬ (a0 = asing a1)) → ain a1 (asm (apow a2) (asing a2)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a2 (apow (asing a1))) → (¬ ain a1 (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (asing a1) a3) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain a2 (asing a3)) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a1 (aun (asing (asing a2)) a4) → ain (apow a2) (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (asm a3 (asing a2) = a2) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 a2) → (¬ aal5 a4) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal2 (asm (apow a2) (apow (asing a1))) → aal2 (asm a4 a2) → aal4 (aun a3 (asing (apow a2))) → aex2 (asm a3 (asing a1)) → aex2 (aun (asing a3) (asing (apow a2))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → aex4 (aun a3 (asing (asing a2)))
In Proofgold the corresponding term root is 7092bb... and proposition id is fb315a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZZ7tqwXzfRD2iuyvAUyWE5c42j61GtpDN)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (¬ ain a0 a0) → (¬ (a1 = a3)) → ain a0 (apow (asing a1)) → (¬ ain a1 (apow (asing a1))) → (¬ (a0 = apow (asing a1))) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain a3 (asing a1)) → (¬ ain (apow a2) a2) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (asing a2) a3) → (¬ ain (apow a2) a3) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain a3 (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal2 (asing a1)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal5 (aun (asing (asing (asing a1))) a4) → aex3 a3 → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex5 (aun a4 (asing (apow a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25)))))
In Proofgold the corresponding term root is 293a6f... and proposition id is 958513...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHYA29eXqt9rCfcth2hNYiPacBTZSUhG8o)
(∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ (a2 = a3)) → ain (asing a1) (asing (asing a1)) → (¬ ain a0 (asing a2)) → ain a1 (apow a2) → (¬ (a1 = asing a1)) → ain (asing a1) (apow (asing a1)) → (¬ ain a3 (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (apow a2) a3) → (¬ (asing (asing a1) = a3)) → (¬ (a3 = asm (apow a2) a2)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a1 (asing a3)) → (¬ ain (asing a2) (asing a3)) → ain a3 (asm a4 (asing a1)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a1 (asm a2 (asing a1)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 a2) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 a3) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 a2 → aex2 (asm (apow a2) a2) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm a4 (asing a2)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a0)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4)
In Proofgold the corresponding term root is 99fed4... and proposition id is 2be0fe...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMU1VRbEKFqJWModknmwXem9QbKHVM5gVip)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (¬ (a0 = a3)) → ain a0 (apow (asing a1)) → (¬ ain a0 (asing a1)) → (¬ ain (asing a2) a1) → asubq a1 (asm a3 (asing a1)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ ain (apow a2) (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain a0 (asing (apow (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) a4) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (asing (apow a2)) → ain a2 (aun a3 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq a3 (asm a4 (asing a3)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → aal3 (asm a4 (asing a2)) → aal4 (aun a3 (asing (asing a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (aun a1 (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25)))
In Proofgold the corresponding term root is b530ff... and proposition id is 96908a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNPFsSfu2g1odxJxrzm2dVMuKqK6LWjyNg)
(∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing a2 = asm a3 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (asm a3 (asing a1) = a3)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ (a3 = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a1 (apow (apow (asing a1))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a1 (apow (asm a3 (asing a1))) → ain (asing a2) (aun (asing (asing a2)) a4) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing a2)) a4) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → aex2 a2 → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))
In Proofgold the corresponding term root is ebf06e... and proposition id is ecc4e7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTYMwzJUqvsCDYB1wbRLjzB6gcW3BbT9yE)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain a1 (apow (asing a1))) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = a3)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asing a2) (asing a3)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a1 (aun a3 (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq a3 (asm a4 (asing a3)) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 a3) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (asm a4 (asing a1)) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (asing (asing a2)) a4) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun a3 (asing (asing a2))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 a4 → aex3 (asm a4 (asing a3)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2)
In Proofgold the corresponding term root is 440641... and proposition id is 637241...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdbeePMY3oe8SpYaHVratdRqnb9Pq7kyay)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, aal4 X24 → aal3 X24) → (¬ asubq a2 a1) → (¬ (a1 = a3)) → (¬ (a0 = asing (asing a1))) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asm a3 (asing a1))) → (¬ (asing a1 = a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ asubq (asing a1) (asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain a3 (asing a2)) → (¬ (a2 = apow a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ (a0 = apow a2)) → (¬ (asing (asing a1) = a3)) → (¬ (asm a3 (asing a1) = a3)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a0 (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun (asing (asing a1)) a4) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (asm (apow a2) a2) → aex3 (aun a2 (asing (apow a2))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2)
In Proofgold the corresponding term root is 6b45ee... and proposition id is 6a0a6b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFKNdmWFapi2H1tPQnEDPdYHNfcN5PDpvf)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a0 a2 → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a1 a0) → asubq a1 a2 → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (aun (asing a1) (asing (asing a1))) → asubq a1 (asm a3 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (asing a1) (asing a1)) → (¬ (a1 = apow (asing a1))) → (¬ ain (asing a2) a2) → (¬ asubq (asing a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain a3 (apow a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (asm a3 (asing a1) = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain a3 (aun a1 (asing a3)) → ain a3 (asm a4 (asing a1)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 a3) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 a3 → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun (asing a2) (apow (asing a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (apow a2) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing (asing a1))) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1))))
In Proofgold the corresponding term root is c8cfc4... and proposition id is 49c3af...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQ1MnnwwDgeJMCUKRmqs44YkwyFXRHB6Kv)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a0 a3 → ain a1 a4 → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain a0 (asing a1)) → ain a1 (apow a2) → (¬ ain a1 (apow (asing a2))) → ain a2 (apow a2) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain a3 (asing a1)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ asubq (asing a2) a2) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ (asing a1 = apow a2)) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (apow a2) (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain a0 (asing a3)) → ain a3 (aun a1 (asing a3)) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm a3 (asing a2) = a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (asm a4 a2)) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun (asing (apow (asing a1))) a4) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) a4
In Proofgold the corresponding term root is 35d511... and proposition id is e01fcc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMX2SULF7HBSFoAYgQmV32fazExxEPko64X)
ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a3 a3) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a0 (apow a2) → (¬ asubq a1 (asing a1)) → (¬ ain (asing a1) a2) → (¬ ain (asing a1) a1) → (¬ (a0 = asing (asing a1))) → (¬ ain a0 (asing (asing a1))) → asubq a1 (asm a3 (asing a1)) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a2 (asm a4 (apow (asing a2))) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm a2 (asing a0)) (asing a1) → (asm a3 (asing a2) = a2) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq a2 (apow (asm a3 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → aal5 (aun (asing (asing a2)) a4)
In Proofgold the corresponding term root is 9e4696... and proposition id is 98d922...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQmg4YkzJWeRBkuYz2phLDDqqTUSWkLE7y)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a0 a2 → ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → ain a1 (apow a2) → ain a0 (apow (asing a2)) → (¬ ain (apow a2) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ asubq a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (asing (asing a1) = a3)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asing a2) (apow (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain (asing a2) (asing a3)) → ain a0 (aun a1 (asing a3)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a1 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (asm a2 (asing a1) = a1) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (asm a4 (asing a3) = a3) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 a4) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aun a2 (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ (a2 = a3))
In Proofgold the corresponding term root is 5a6afd... and proposition id is 59cc97...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXUsaeHjGJursPKHXiSzF4g6LmjuiydLQM)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a1 a1) → (¬ ain a3 a2) → ain (asing a1) (apow a2) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain (asing a1) (asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ (asing a1 = asing a2)) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (asing a2 = apow a2)) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a3 (asm a4 (asing a2)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (asing (apow a2)) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a1)) a1 → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal2 (asing a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal4 a4 → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (asing a2))) → aex4 a4 → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq a2 (aun (asing a1) (apow (asing a2)))
In Proofgold the corresponding term root is de365f... and proposition id is 762b90...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLDsWHYRtA2jwaU3YjvuRZBwdZR4sYxFew)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (¬ asubq a3 a2) → (¬ (a1 = a2)) → (¬ (a1 = a3)) → ain a1 (asm (apow a2) (apow (asing a1))) → ain a2 (asm a3 (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain (asing a2) a2) → (¬ (asing a1 = asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (asing a2 = apow a2)) → ain a0 (apow (asing (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a2) a4) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a0 (asing a3)) → ain a3 (asing a3) → (¬ ain a2 (aun a1 (asing a3))) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a3 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (asm (apow a2) (asing a1)) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun (asing (asing a1)) a4) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (apow a2) (asing a0)) → aex5 (aun (asing (asing a1)) a4) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 457c1e... and proposition id is f6504a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZVXj5kwyMtxL3VhHRN8icXKApKG6eUpWt)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, aal4 X24 → aal3 X24) → (¬ ain a1 a1) → (¬ ain a2 a1) → asubq a1 a2 → (¬ asubq a3 a2) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (¬ ain a1 (asing a2)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a2 (apow a2) → (¬ ain a0 (asm (apow a2) a2)) → (¬ asubq (asing a1) (asing a2)) → (¬ (asing a1 = apow a2)) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (apow (asing a1) = a3)) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a1 (apow (asm a3 (asing a1))) → ain a0 (aun a1 (asing a3)) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a0 (aun a1 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq a1 (asm a2 (asing a1)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal2 (asing a3)) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 (asm (apow a2) a2) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → ain a0 (asm (apow a2) (asing a1))
In Proofgold the corresponding term root is 30ad5f... and proposition id is 0e5f85...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVWyf6LjPoKBKqEj9k6W6cDb21qZZRsLKe)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → asubq a3 a4 → (¬ (a0 = a1)) → (¬ asubq a1 (asing a1)) → (¬ asubq a2 (asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) a3) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a3)) a3 → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (¬ aal5 a4) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (aun a2 (asing (apow a2))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex5 (aun a4 (asing (apow a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25)
In Proofgold the corresponding term root is 32cd81... and proposition id is 7f0ba8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRZGrkkc43pzbwXRnQwevXLwuEow8emepm)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a1 a0) → (¬ ain (asing a1) a0) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) a2) → (¬ ain (asing a1) a1) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain (asing a1) a3) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain a0 (apow (asing (asing a1))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a2 (asm a4 (asing a1)) → (¬ ain a1 (asing a3)) → ain a3 (asm a4 (asing a2)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a3 (aun (asing (asing a2)) a4) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 a2) (asing a2) = asing a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 a3) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal3 (asm (apow a2) (asing a2)) → aal3 a3 → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (aun (asing a2) (apow (asing a2))) → aex2 (asm a3 (asing a1)) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ (a1 = apow a2))
In Proofgold the corresponding term root is 599ff8... and proposition id is 7088ab...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMM4B6mA9tYVtDZMfafij1bDaTRA2FAaNAV)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → asubq a1 (asing a0) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (¬ ain a2 (apow (asing a1))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a2 (asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (a2 = apow a2)) → (¬ ain (asing (asing a1)) a3) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a2 (asing a0)) (asing a1) → asubq a1 (asm a2 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a0)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal2 (apow (asing (asing a1))) → aal3 (asm a4 (asing a1)) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 33bc58... and proposition id is 0ebf32...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVn47YpfpzV9gtADSYDbradHahyjpraNFi)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → asubq a2 a3 → asubq a3 a4 → (¬ (a0 = a1)) → (¬ (asing a1 = apow a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ ain (apow a2) (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (asm a3 (asing a1) = apow a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (asm a4 (asing a1)) → ain a2 (asm a4 (apow (asing a2))) → ain a3 (asm a4 (apow (asing a2))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (asm a3 (asing a2) = a2) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow a2) (asing a1)) → aal4 (apow a2) → aal4 a4 → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (apow a2) (asing (apow a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is e03856... and proposition id is 3482c0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMddp1noRVXy6FqBdphimGDjTfSbNLhcPp2)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ asubq a3 a2) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain (asing a1) a0) → ain a0 (asm a3 (asing a1)) → ain (asing a1) (apow a2) → (¬ ain (asing a1) (asing a2)) → ain a2 (asm a3 (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asm (apow a2) a2) a2) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain a0 (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (asing a3)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (asm a2 (asing a0) = asing a1) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 a2) (asing a3) = asing a2) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal2 (aun (asing a1) (asing (asing a1))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a1)) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ ain (asing a2) (asing (asing a1)))
In Proofgold the corresponding term root is 46500a... and proposition id is 5ae410...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMX8pdWBnZ4rHwkux2cJ1RDbahuyygVyHit)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a0 a0) → (¬ (a0 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a0 (apow a2) → (¬ (a0 = asing (asing a1))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ (a2 = asing a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) a2)) → ain a0 (apow (apow (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain a0 (asm a4 a2)) → ain a2 (asm a4 a2) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal4 (apow a2) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is a725f6... and proposition id is 32d518...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMViZw567XBkgMHejGXJENkqqmoA87ZLuJH)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain (asing a1) a0) → ain a0 (apow (asing a1)) → (¬ ain a1 (apow (asing a1))) → (¬ ain a0 (asing (asing a1))) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (asing a1 = asm (apow a2) a2)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain a3 (apow a2)) → (¬ (asing (asing a1) = a3)) → (¬ (asm a3 (asing a1) = a3)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun a3 (asing (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asing (asing a1)) a4) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq a1 (asm a2 (asing a1)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (asm a4 (asing a1)) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (asm a4 (asing a0) = asm a4 (apow (asing a2)))
In Proofgold the corresponding term root is 69119e... and proposition id is 6e8a53...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZsB14diYFjGZk13c45onW6rgWpMuT5kwd)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain a0 (asm (apow a2) (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain a3 (asing a2)) → (¬ ain (apow a2) (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm a4 (asing a1)) → ain a3 (asm a4 (apow (asing a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → asubq (asm a3 (asing a2)) a2 → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 (asm (apow a2) (asing a1)) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (apow a2) (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → ain a0 a3
In Proofgold the corresponding term root is 7be2d1... and proposition id is dc09b2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSwxCf9N1yJd4z9TjQYY9AcQFJebUrMnYR)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → ain a0 (asm (apow a2) (asing a2)) → ain a0 (apow (asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a2 = apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → asubq a3 (apow a2) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (a3 = apow a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (apow (asm a3 (asing a1))) → ain a3 (aun a1 (asing a3)) → ain a3 (aun (asing a1) (asing a3)) → ain a3 (asm a4 (asing a1)) → ain a0 (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → aal2 (asm a3 (asing a1)) → aal2 (asm (apow a2) a2) → aal3 (asm a4 (asing a1)) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 a2 → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (asm a4 (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2))))
In Proofgold the corresponding term root is 2abd6e... and proposition id is fa364e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVAEffKxUwMu9DkBtWMuu6bF2d14Ad9Vme)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → ain a1 (asing a1) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain a0 (asing (asing a1))) → (¬ (a1 = asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (¬ (a2 = apow a2)) → (¬ ain (apow a2) (asing a2)) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing (asing a1)) a4) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 a2 → aal3 (asm (apow a2) (asing a2)) → aal3 (aun (asing a2) (apow (asing a2))) → aal3 (asm a4 (asing a2)) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm a4 (asing a2)) → aex4 (aun a3 (asing (apow (asing a1)))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ asubq (asm a3 (asing a1)) (asing a2))
In Proofgold the corresponding term root is ea8f3d... and proposition id is 62e0f5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNcpo8vBQhmYesWJQPGg85zhgJacJvFWdT)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → asubq a3 a4 → (¬ (a0 = asing (asing a1))) → ain a1 (asm (apow a2) (asing a2)) → (¬ (asing a1 = a2)) → asubq a1 (asm a3 (asing a1)) → (¬ ain (asing a1) (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ (a2 = apow a2)) → (¬ (a1 = asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a2 (asm a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal2 (asm a4 a2) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal5 (aun (asing (asing a1)) a4) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex5 (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (¬ aal4 (asm a4 (asing a2)))
In Proofgold the corresponding term root is 13f4b7... and proposition id is 1f1921...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQarKJVAbZMjNLYPgv2Y52QJR1K1CYcDBz)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a2 a1) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain (asing a1) a1) → ain a2 (asm a3 (asing a1)) → ain a2 (apow a2) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain a1 (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain (asing (asing a1)) a4) → (¬ ain a0 (asm a4 a2)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a0 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm a4 (asing a3)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq a3 (aun a3 (asing (asing a2)))
In Proofgold the corresponding term root is 3490d0... and proposition id is 072717...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZHQkeK62DAEwxazTVXr4jF9c61QufRob7)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a0 (asm a3 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ (asing a1 = asing a2)) → (¬ ain (asing a1) a3) → (¬ ain a3 (apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (asing a2 = a3)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (asm (apow a2) a2 = apow a2)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain a2 (asm a4 (asing a1)) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain (apow (asing a1)) a4) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 a2 → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ ain a3 a3)
In Proofgold the corresponding term root is c95195... and proposition id is 9250ba...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZ8gjAn3i959xrxqdML2DzRWBQztRHGkmw)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a2 a2) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → asubq a1 (apow (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (asing a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (apow a2) a3) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain a1 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a2)) a2 → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal2 a2 → aal2 (asm (apow a2) (apow (asing a1))) → aal4 (apow a2) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 a2 → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm a4 (asing a3)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))))
In Proofgold the corresponding term root is c669f0... and proposition id is 5fcf51...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVqzkaq2xXEN38d1aPct3wWJQLR5tdZhWz)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (¬ ain a3 a3) → asubq a2 a3 → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asing a2)) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a2) (apow a2)) → (¬ ain (apow a2) (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (asing a2 = a3)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → (¬ ain a0 (asing a3)) → ain a1 (asm a4 (asing a2)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun a4 (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq a3 (asm a4 (asing a3)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal2 (asing (apow a2))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → aal2 (asm (apow a2) a2) → aal3 (aun (asing a2) (apow (asing a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm (apow a2) a2) → aex2 (asm a4 a2) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 a4 → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2))))
In Proofgold the corresponding term root is 51fc1b... and proposition id is 97220a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYvuR8b4Co9CWkY9z9kfh1FNb8qNBrLBow)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a2 (asing a2) → ain a2 (apow a2) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asing a2)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain a3 (asm a3 (asing a1))) → ain a1 (apow (apow (asing a1))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 (apow (asing a1))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 a3 → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun a1 (asing (apow a2))) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (¬ ain a3 (asing a2))
In Proofgold the corresponding term root is 0f77f1... and proposition id is 833e12...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLStnfiR4eyKzuPCNHNmEkUMxTvgnqzZXN)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain (asing a1) a0) → (¬ ain a0 (asing a1)) → ain a2 (asm a3 (asing a1)) → (¬ (a1 = apow (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (a1 = apow a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → asubq a3 (apow a2) → (¬ (a3 = asm (apow a2) a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ ain a3 (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain a2 (asm a4 (asing a1)) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain a2 (asing a3)) → ain a3 (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a1 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a2 (asing a0) = asing a1) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 (apow (asing (asing a1))) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex5 (aun (asing (apow (asing a1))) a4) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (asm (asm a3 (asing a1)) (asing a0) = asing a2)
In Proofgold the corresponding term root is bea551... and proposition id is f131fb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbi4Dc5txS8JNazEzfi3PJH5tvSAp4oZhr)
ain a0 a2 → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a3 a3) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ ain a1 (asing a2)) → ain (asing a1) (apow (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm (apow a2) (apow (asing a2))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ (a2 = asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a2) a4) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a3 (asm a4 (asing a1)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 a2) (asing a2) = asing a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 (asm (apow a2) a2) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (asm (apow a2) a2) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (aun (asing a2) (apow (asing a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4)
In Proofgold the corresponding term root is af77cb... and proposition id is b13a73...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUs3CRo8QUsHqCtBBTpyS53wz89TWRxQhc)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → ain a1 (asm (apow a2) (apow (asing a1))) → asubq (asing a1) a2 → (¬ ain a1 (asing a2)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (apow a2) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ (asing a2 = a3)) → (¬ ain (asing a2) (asm (apow a2) a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal2 a1) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 (asm a4 a2) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2)
In Proofgold the corresponding term root is ba040f... and proposition id is fd50a6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbbd9UsHvdQ9VjM69uwGsGj7bcnaNMRn2g)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a0 a2 → ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ ain a0 (asing a2)) → (¬ (a1 = asing a1)) → (¬ ain a1 (apow (asing a2))) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm (apow a2) (asing a1)) → (¬ (a1 = asing (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ (a2 = asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain a0 (asing a3)) → ain a3 (aun a1 (asing a3)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a0 (aun a2 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal5 (aun (asing (apow (asing a1))) a4) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm a3 (asing a1)) → aex3 (asm (apow a2) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24))))
In Proofgold the corresponding term root is 5d8533... and proposition id is 1cc0a8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZCmVUuPuXVpsAmHax67m7vU7syUWryXG7)
(∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → asubq a3 a4 → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a0 (asm a3 (asing a1)) → asubq (asing a1) a2 → (¬ (a0 = asm (apow a2) a2)) → ain a2 (asm a3 (asing a1)) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain a1 (asing (asing a1))) → (¬ (asing a1 = a3)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ ain (apow (asing a1)) a3) → (¬ ain (asm (apow a2) a2) a3) → (¬ (asing a2 = asm (apow a2) a2)) → ain a0 (apow (asing (asing a1))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (apow a2) (aun a2 (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a2 (asing a0) = asing a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → (¬ aal3 a2) → (¬ aal3 (asm a3 (asing a1))) → aal2 a2 → aal3 (asm a4 (asing a2)) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm (apow a2) a2) → aex3 (asm a4 (asing a0)) → aex3 (asm a4 (asing a3)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24)
In Proofgold the corresponding term root is f73418... and proposition id is 0f3b98...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQCt3M6YzNowkkgoskTM8dUpYUJzfFrnpi)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → asubq (asing a1) a2 → (¬ ain (asing a1) a1) → (¬ ain a1 (asing a2)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → (¬ aal2 (asing (asing a1))) → (¬ aal2 (asing a3)) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 a4) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal5 (aun (asing (asing (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aun a1 (asing a3)) → aex3 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing (asing a1))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → ain a1 (aun a3 (asing (asing a2)))
In Proofgold the corresponding term root is 3168a1... and proposition id is 4dcd8f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMS6shXQvDPW2cwzt4vQ8bKC71osnzpK7jG)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (¬ ain a2 a2) → (¬ asubq a3 a2) → (¬ asubq a4 a3) → (¬ (a0 = a1)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ (a0 = apow (asing a1))) → ain a2 (asm a3 (asing a1)) → ain a2 (asm (apow a2) a2) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a2 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → aal4 (apow a2) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a3))
In Proofgold the corresponding term root is 2f1c5c... and proposition id is f07559...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFhPAFPkVEoXK1PVDTRkDkLKgZ1kULAbvj)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ ain a1 a1) → (¬ (a2 = a3)) → (¬ ain a1 (apow (asing a1))) → ain a0 (asm (apow a2) (asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) (asing a1)) → (¬ ain a3 (asing a1)) → (¬ ain (asing a2) a2) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ ain (asing (asing a1)) a3) → (¬ ain (asm (apow a2) a2) a3) → asubq (asm a3 (asing a1)) (apow a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (a3 = asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (apow a2) (asing (asing a1)) = a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm a4 a2) → aal3 (asm (apow a2) (asing a2)) → aal4 a4 → aex2 (apow (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is cf756f... and proposition id is 97e5b1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWtkAL3KsfqfgwZJHPFnFRr6fU5iqmosa7)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a2 a3 → ain a0 a4 → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (¬ asubq a2 a1) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a1 (apow a2) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asm (apow a2) a2)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a2 = asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → (¬ ain (asing a2) (asing a3)) → (¬ ain a2 (aun a1 (asing a3))) → ain a1 (aun a2 (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 a3) → (¬ aal5 a4) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal3 a3 → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (apow a2) (asing (asing a2))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (asm a4 (asing a3) = a3)
In Proofgold the corresponding term root is 39e2d4... and proposition id is a4fbc6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVdzw3BkRf35EqU483ES4VeiHycS6Gg32C)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ asubq a3 a2) → (¬ ain a1 (asing a2)) → ain a1 (asm (apow a2) (asing a2)) → (¬ (a0 = asm (apow a2) a2)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ asubq a2 (asing a2)) → (¬ (a1 = asing (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (a0 = apow a2)) → asubq a3 (apow a2) → (¬ ain (asing a2) (asm (apow a2) a2)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (a3 = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a3 (aun (asing a1) (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal3 (asm a4 a2)) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (asing (asing a2)) a4) → aex2 (asm a4 a2) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex5 (aun (apow a2) (asing (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) a4)
In Proofgold the corresponding term root is ca886e... and proposition id is 759b03...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPn3MSCwTJMXyCNAi72Us5fu1aSJPWv7yx)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a0 a2 → ain a2 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ asubq a2 a1) → (¬ asubq a1 a0) → (¬ (a0 = asing (asing a1))) → (¬ (a0 = asm (apow a2) a2)) → ain a2 (asm (apow a2) a2) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ (asm a3 (asing a1) = a3)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain a0 (asing a3)) → (¬ ain a2 (aun a1 (asing a3))) → ain a1 (aun (asing a1) (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm a4 (apow (asing a2))) → ain a1 (aun (asing (asing a2)) a4) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a0 (aun a1 (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm a4 (asing a3)) a3 → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal5 (aun (apow a2) (asing (apow a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 a2 → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (apow (asing a1))) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ ain (asm a3 (asing a1)) a2)
In Proofgold the corresponding term root is 5666d3... and proposition id is 0d0850...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMEqkucaShRJWeoQSQzUJ91UTx7eqjGd1xL)
(∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (¬ ain a1 (apow (asing a1))) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain a0 (asm (apow a2) a2)) → ain a2 (asm (apow a2) (asing a1)) → (¬ (a1 = asing a2)) → (¬ ain (apow a2) a2) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) a3) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a2 (asm a4 a2) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a3 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq a3 (asm a4 (asing a3)) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal2 (asm (apow a2) a2) → aal2 (asm a4 a2) → aal3 (asm (apow a2) (apow (asing a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun a1 (asing a3)) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ ain a3 (apow (asing a2)))
In Proofgold the corresponding term root is 1731a9... and proposition id is 986064...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHGCKx7BXiAcUV7U1bkR9QBECJVE9ZWBiP)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (¬ ain X24 a0)) → ain a0 a2 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ (a1 = asing a2)) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = a3)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain a3 (apow a2)) → (¬ ain (asing (asing a1)) a3) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a2 (asm a4 (asing a1)) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a0 (asm a4 (asing a2)) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 a4) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (aun a2 (asing (apow a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain (asing a1) (asm (apow a2) (asing a1))
In Proofgold the corresponding term root is 337ed3... and proposition id is cdf724...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUkoU2dAhrdnsrTqdXuWZHk4awwi9GvHAC)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ ain a2 (asing a1)) → ain a2 (asm (apow a2) a2) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asing a2) a3) → (¬ (a1 = apow a2)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun a3 (asing (asing a2))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a3 (asm a4 (asing a2)) → ain a3 (aun (asing (asing a2)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a1 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq a4 (aun a4 (asing (apow a2))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → (asm (asm a3 (asing a1)) (asing a2) = a1) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (apow (asing a1)) → aex2 (asm (apow a2) a2) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun a3 (asing (apow a2))) → aex5 (aun (asing (apow (asing a1))) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is 153339... and proposition id is 5d3695...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMddr7Vb5VgRrGostbMKH6NqmP8QsHYLNLk)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a2 (apow a2) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a2) a2) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain a3 (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → ain a3 (asm a4 (asing a2)) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a0 (aun (asing (asing a2)) a4) → ain a0 (aun a2 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (asing a3))
In Proofgold the corresponding term root is 1fb0ee... and proposition id is aea6ef...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMF99aZjKeCCjPSVT3gWWPEaToZfiE1J6F7)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (¬ asubq a2 a1) → (¬ ain a0 (asing a2)) → (¬ (a0 = apow (asing a1))) → ain a0 (apow (asing a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asm (apow a2) a2) a3) → (¬ (a1 = apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (asm a3 (asing a1) = apow a2)) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a2 (asm a4 (asing a1)) → (¬ ain (asm a3 (asing a1)) a4) → ain a2 (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm a2 (asing a0)) (asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (asm a3 (asing a1)) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 7093cc... and proposition id is d7a6f0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHAXRrPhW5tWvE63Zt61aYYD7LL41Gi4ew)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ asubq a2 a1) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain (asing a1) (asing (asing a1)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a1 = asing (asing a1))) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm (apow a2) a2)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ ain a1 (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a3 (asm a4 a2) → ain a0 (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal2 a2 → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (asm (apow a2) (apow (asing a1))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex5 (aun (asing (asing (asing a1))) a4) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25))
In Proofgold the corresponding term root is 67a4ea... and proposition id is 16d209...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXSiTJEP25mZWsye74Mmg99GegLY6ukcLf)
(∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ ain a0 (asing a2)) → (¬ (a0 = apow (asing a1))) → (¬ ain (asing a2) a1) → ain a2 (apow a2) → (¬ ain (apow a2) a2) → (¬ (a2 = asm (apow a2) a2)) → (¬ (a0 = apow a2)) → asubq a3 (apow a2) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain (asm (apow a2) a2) a4) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a3 (asm a4 (apow (asing a2))) → ain a3 (aun (asing (asing a2)) a4) → ain a1 (aun a2 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing a2)) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (apow (asing (asing a1))) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (asing (asing a2)) a4) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun a1 (asing (apow a2))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ ain a3 (apow (asing a2)))
In Proofgold the corresponding term root is 1d67e3... and proposition id is cfc9d0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUhdFDtdDBQsSZ6BW7UHEgan51se6DYMf8)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (asing a2)) → (¬ asubq a1 (asing a2)) → ain a2 (asm (apow a2) a2) → ain a2 (asm (apow a2) (asing a1)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a2) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ ain (apow a2) (apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a0 (asm a4 a2)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a3 (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal3 a3 → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (asm a4 (asing a2)) → aal5 (aun (apow a2) (asing (apow a2))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex5 (aun (asing (asing a1)) a4) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1))))
In Proofgold the corresponding term root is 8a20a6... and proposition id is 7c265f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYd3pT4kLSyHV2HEPiXiwCorChqEZwk13J)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → asubq a1 (asing a0) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (¬ asubq a2 a1) → (¬ ain a2 (apow (asing a1))) → (¬ ain a2 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain a3 (apow a2)) → (¬ (asm a3 (asing a1) = a3)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ (asing a2 = apow a2)) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain a3 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow a2) (asing a2)) → aex2 a2 → aex2 (asm (apow a2) a2) → aex2 (apow (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3))
In Proofgold the corresponding term root is 7d9ec1... and proposition id is ece01a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMHFwV1tcuGWKHQ287s8dxSGaJHULytwWC4)
ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → ain a2 (asing a2) → (¬ (a0 = asing a1)) → ain (asing a1) (asm (apow a2) a2) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (asing a1 = apow a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ (apow (asing a1) = a3)) → ain a1 (apow (apow (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain a1 (apow (asm a3 (asing a1))) → ain a0 (aun a1 (asing a3)) → ain a3 (asm a4 (asing a2)) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal2 (asm a4 a2) → aal5 (aun (asing (apow (asing a1))) a4) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 a2 → aex2 (aun a1 (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → (¬ ain (asing a2) a1)
In Proofgold the corresponding term root is 99fedb... and proposition id is cc3882...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUTpau7zmwbvCou85Y51x9Y995ajFD1Sn4)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a1 a3 → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → asubq a3 a4 → (¬ asubq a2 a0) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a0 (apow (asing a1)) → (¬ ain a1 (apow (asing a2))) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a2))) → (¬ ain a1 (asing (asing a1))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (a1 = apow a2)) → (¬ (apow (asing a1) = a3)) → (¬ (asing a2 = asm (apow a2) a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a0 (asm a4 a2)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a3 (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain a0 (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 a3) → (¬ aal4 (asm a4 (apow (asing a2)))) → aal2 a2 → aal2 (aun (asing a1) (asing (asing a1))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 a3 → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (apow a2) → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24))))
In Proofgold the corresponding term root is c2ec55... and proposition id is ce2b16...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFyqcBra9vaxaCqw59wdXXugkfqgiDm8yj)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → ain a0 a4 → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a2 a2) → (¬ ain a3 a2) → (¬ asubq a2 a1) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ asubq a2 (asing a2)) → (¬ ain (asing a1) (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ asubq a3 (asing a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain (asing a1) a4) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain a3 (asing a3) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asing a2) (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal2 a2 → aal3 a3 → aal3 (asm a4 (asing a2)) → aal5 (aun (asing (asing a1)) a4) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 a2 → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (asm (apow a2) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (apow a2) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → ain (asing a2) (aun (apow (asing a2)) (asing a3))
In Proofgold the corresponding term root is 84f72d... and proposition id is a3603b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbVTA74sEBqo1EDiKkuLLYnovnaPH7E1Di)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a1 a3 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ asubq a1 a0) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a1 (apow a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain a2 (apow a2) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (asing (asing a2)) → ain (asing a2) (apow (asing a2)) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a0 (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2
In Proofgold the corresponding term root is 58b633... and proposition id is 2052d2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFcLMoS7imZKN4q4bt6K6sMsNJrfmzdNSX)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → ain a0 a2 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → asubq a1 (apow (asing a1)) → (¬ asubq a2 (asing a1)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ asubq (asing a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ (apow (asing a1) = a3)) → (¬ (asing a2 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ (asm (apow a2) a2 = apow a2)) → ain a1 (apow (apow (asing a1))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a2) (asing a3)) → ain a1 (asm a4 (asing a2)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asing a1) (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing (apow a2))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → aal2 (asm a4 a2) → aal4 a4 → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (apow (asing a1)) → aex2 (asm (apow a2) a2) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2))
In Proofgold the corresponding term root is c6963a... and proposition id is efa592...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMb5jdcCTLSrzYvqL42wGmEh9QrngTYu3Vt)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ (a0 = a1)) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a0 (apow (asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a1) a1) → (¬ ain (asing a1) (asing a2)) → (¬ asubq a2 (asing a1)) → (¬ ain a2 (apow (asing a2))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain (apow a2) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ (apow (asing a1) = a3)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a3 (aun (asing a1) (asing a3)) → ain a1 (asm a4 (asing a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (asing a2) (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun a3 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asm a2 (asing a0)) (asing a1) → asubq a1 (asm a2 (asing a1)) → asubq (asm a3 (asing a2)) a2 → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal3 (asm a4 (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 a4 → aex3 (asm a4 (asing a0)) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → ain a1 (aun (asing a1) (asing (asing a1)))
In Proofgold the corresponding term root is 5ec604... and proposition id is 8900a8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TML6ugTR6aCJH8xASP7BMLbaa1bxF1TcpQg)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (¬ ain a1 a0) → (¬ ain a1 a1) → (¬ ain a3 a2) → ain (asing a1) (asing (asing a1)) → ain a1 (apow a2) → (¬ (a0 = apow (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain (apow a2) a2) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (asing a1) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a2 (asm a4 a2) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asing (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm a3 (asing a2) = a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq a3 (asm a4 (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 (asm (apow a2) a2) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 a4 → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (asm (apow a2) a2) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun a3 (asing (asing a2))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1))))
In Proofgold the corresponding term root is 1b66ac... and proposition id is 6a8c36...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMaUzgyLkvxbQeKN8fd7A4pRcXqn5VHLFEd)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → (¬ (a0 = a1)) → ain (asing a1) (asing (asing a1)) → ain a2 (asing a2) → (¬ (asing a1 = apow a2)) → (¬ (a2 = asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a1 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (asm (apow a2) a2) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → aex3 (asm a4 (asing a3))
In Proofgold the corresponding term root is 95d246... and proposition id is 0a4828...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUbamG2BXwK6A3rMAbLUqwvrLG3fLibLYU)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal5 X24) → (¬ aal6 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → ain (asing a1) (apow a2) → (¬ ain a1 (asing a2)) → (¬ ain a1 (asm a3 (asing a1))) → (¬ (a1 = apow (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ ain (asing a2) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (asing a2) a3) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain (apow a2) a4) → (¬ ain (asing a1) (asm a4 a2)) → ain a1 (aun (asing (asing a2)) a4) → ain a0 (aun a3 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm a2 (asing a0)) (asing a1) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → aal5 (aun (asing (asing (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1))))
In Proofgold the corresponding term root is 70da68... and proposition id is 041352...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSkMi8NwNWRQ5UnjWcDZfDgwoxiDHgztCV)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → asubq a2 a3 → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain a0 (asing a1)) → ain (asing a1) (asm (apow a2) a2) → (¬ ain a2 (apow (asing a2))) → ain a2 (asm (apow a2) (apow (asing a2))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = a3)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → (¬ (asing a2 = apow a2)) → (¬ ain a0 (asing (apow (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain a0 (aun a1 (asing a3)) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq a2 (apow (asm a3 (asing a1))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25))
In Proofgold the corresponding term root is 9a87b2... and proposition id is a7ffd6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMN9FXfwBTxeZryeAchtwocnZewsPwGbmR)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a3 a3) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a1 (apow a2) → ain a1 (asm (apow a2) (apow (asing a1))) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (apow (asing a1)) a3) → (¬ ain (asm (apow a2) a2) a3) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (apow (apow (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a1 (asm a4 (asing a2)) → (¬ ain (asing (asing a1)) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun a4 (asing (apow a2))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (¬ aal6 (aun a4 (asing (apow a2)))) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (apow (asing a2)) → aex2 (aun (asing (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ (a2 = asing (asing a1)))
In Proofgold the corresponding term root is 864a37... and proposition id is 4ac721...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYTfc8VJJM1mstUnjZStDyAbpiUf39KNkV)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (¬ ain X24 a0)) → ain a0 a3 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (¬ (a0 = asing (asing a1))) → (¬ ain a1 (asing (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ ain (asing a2) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (apow a2) a3) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (apow a2) (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain a1 (asing a3)) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (asing a1) (asm a2 (asing a0)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal2 a1) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))
In Proofgold the corresponding term root is e8f9a3... and proposition id is db0c92...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMDDXNascYjJZp14GGBzwX3mDMkjQ8Lr9Y)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a3 a3) → (¬ asubq a1 a0) → (¬ (a0 = asing a1)) → (¬ (a0 = asing a2)) → (¬ (a0 = asm (apow a2) a2)) → (¬ asubq (asing a1) (asing a2)) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (a2 = apow a2)) → (¬ ain (asing a2) a3) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a0 (asm a4 (asing a1)) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a2)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a1 (aun (asing (asing a2)) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm a3 (asing a2)) a2 → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq a3 (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal6 (aun a4 (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal2 (asm (apow a2) (apow (asing a1))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aal5 (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1)))
In Proofgold the corresponding term root is 55f25e... and proposition id is e342ea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZH74f6LBPJbDgUr2KmfyK3z1De6os4Pxy)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ (a1 = a2)) → (¬ asubq a2 a0) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → ain a0 (apow a2) → (¬ ain a2 (apow (asing a2))) → (¬ asubq (asing a1) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asing a1) a4) → ain a3 (asm a4 (asing a2)) → (¬ ain (asm (apow a2) a2) a4) → ain a2 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq a1 (asm a2 (asing a1)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (aun a1 (asing a3)) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (aun a3 (asing (apow a2))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq a1 (asing a1))
In Proofgold the corresponding term root is f09287... and proposition id is 6ee985...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPBUWFw7axZGCzyJcUQgPWpuCu7qubuppS)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → ain a1 a2 → ain a0 a3 → ain a2 a3 → ain a1 a4 → ain a3 a4 → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (¬ (a0 = asing a1)) → (¬ asubq a1 (asing a2)) → (¬ ain (asing a1) a1) → ain a2 (asm a3 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = asing a2)) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain a3 (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain (asing a2) a4) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a3 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain a0 (aun a1 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → aal2 (asm (apow a2) (apow (asing a1))) → aal3 (aun a2 (asing (apow a2))) → aal5 (aun (apow a2) (asing (apow a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm a3 (asing a1)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1))
In Proofgold the corresponding term root is 5cc082... and proposition id is c4b649...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcdwxQeZYFcNfe4VJ5FJGAgBGQiSmo1qAt)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a2 a0) → (¬ ain a3 a3) → (¬ asubq a2 a1) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain (asing a1) (asing (asing a1)) → (¬ ain a1 (apow (asing a1))) → ain a2 (asm (apow a2) a2) → (¬ ain a2 (apow (asing a2))) → (¬ (a1 = asm a3 (asing a1))) → (¬ ain (asing a2) a2) → (¬ ain (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain a3 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (asm (asm a4 a2) (asing a3) = asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 a1) → (¬ aal3 (apow (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex3 (asm a4 (asing a3)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun (apow a2) (asing (asing a2))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ ain a3 (asing a1))
In Proofgold the corresponding term root is 4cb58d... and proposition id is 9fad0e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLpXkpwxtJ3boxgP3WH7pi6QFpRDNUgu9y)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → asubq a2 a3 → (¬ asubq a2 a0) → (¬ asubq (asing a1) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain a0 (asing (apow (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain (asing a2) a4) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (apow (asing a1)) a4) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → ain a3 (aun (asing (asing a2)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 a2 → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 (aun a3 (asing (apow a2))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (apow (asing a2)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1))
In Proofgold the corresponding term root is 332d94... and proposition id is 772344...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQmtP6rp6jdNYMYTxhM1Q8aG5QmTAcbidD)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (¬ (a1 = a3)) → ain a1 (asing a1) → (¬ ain a0 (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain a3 (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (apow (asing a1)) a3) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain (apow a2) a4) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (apow a2)) → aal3 (asm a4 (asing a2)) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (asm a4 (asing a3)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex5 (aun (apow a2) (asing (apow a2))) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ ain (apow (asing a1)) a4)
In Proofgold the corresponding term root is 280545... and proposition id is f0ed84...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNuXVjgAFqBPf4kPfetzfjUHfq8rXcEUTv)
ain a2 a3 → ain a1 a4 → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ (a0 = a1)) → (¬ ain (asing a1) a0) → (¬ ain (asing a1) a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain (asing a2) a1) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a3 (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a2 = asm (apow a2) a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain (asing a2) (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a0 (aun (asing (asing a2)) a4) → (¬ ain a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) a4 → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal4 (aun a3 (asing (apow (asing a1)))) → aex2 (asm a4 a2) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing a0)) → aex3 (asm a4 (asing a0)) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25)))
In Proofgold the corresponding term root is 67eee7... and proposition id is a51994...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWW8CLnk7porA1mw7jHUX7hL5fZpmmmNCv)
(∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (¬ ain a2 a0) → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (asing a1)) → (¬ ain a2 (apow (asing a2))) → ain a2 (asm (apow a2) (apow (asing a2))) → ain a0 (apow (asing a2)) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (apow a2) (asing a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ ain (apow a2) a3) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain (asing a1) (asm a4 a2)) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (asing (apow a2)) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 a3) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal3 (asm a4 (asing a1)) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (asm a3 (asing a1)) → aex2 (asm a4 a2) → aex2 (asm a3 (asing a2)) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25)
In Proofgold the corresponding term root is 0f1b8d... and proposition id is 34e783...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbwkGes35rRRjcy4oJYQJ5qWKs7JYfjUiq)
(∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (¬ ain a2 a0) → (¬ ain a3 a2) → (¬ (a1 = a3)) → (¬ asubq a1 a0) → (¬ ain (asing a2) a1) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asing a2)) → (¬ ain (asing a2) (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (asing a2) a4) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a0 (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a0 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a2)) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal2 (asing (apow a2))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (asm a3 (asing a1)) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aun a1 (asing a3)) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))
In Proofgold the corresponding term root is ea5f2b... and proposition id is 4c491c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbvKNUdNmFUVJXCHpV6PER3ysQwKaJ9xdP)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ ain a1 a0) → ain a0 (asm (apow a2) (asing a2)) → (¬ ain a0 (asing (asing a1))) → (¬ asubq a2 (asing a2)) → (¬ (a1 = asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain (apow a2) (asing a2)) → (¬ (a2 = asing a2)) → (¬ ain (apow a2) a3) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain a0 (apow (apow (asing a1))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asing a2) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a2 (aun a2 (asing (apow a2))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (asm a3 (asing a1))) → aal4 (apow a2) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (asm (apow a2) a2) → aex3 (asm (apow a2) (asing a1)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (apow a2) (asing (apow a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → asubq a3 a4
In Proofgold the corresponding term root is ceaac9... and proposition id is 1f9880...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWz2fuN5e5gkRrMWHxH6SzPWuiDfpFAkK3)
ain a1 a4 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → asubq a1 (asing a0) → asubq (asing a0) a1 → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a0 a0) → (¬ ain a2 a2) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a2) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun a3 (asing (asing a2))) a3) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm a3 (asing a1)) a4 → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → aal3 (asm (apow a2) (asing a2)) → aal3 (aun (asing a1) (apow (asing a2))) → aex2 (asm (apow a2) a2) → aex3 (aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (apow a2) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2)))
In Proofgold the corresponding term root is 395e10... and proposition id is fe5596...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMc6mhDGxo6JdBgNutWKvnJxepNg2VXatbe)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a0 (apow (asing a1)) → (¬ ain a0 (asing a2)) → (¬ (a0 = apow (asing a1))) → (¬ ain (asing a2) a1) → ain a2 (asm (apow a2) (asing a1)) → (¬ (a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a2) a4) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (apow a2) a4) → ain a3 (asm a4 a2) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (asing (apow a2)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 a4) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (apow (asing a1)) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (asm (apow a2) (asing (asing a1))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex3 (asm a4 (asing a0)) → aex5 (aun a4 (asing (apow a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4))
In Proofgold the corresponding term root is ba0a59... and proposition id is b3ec17...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMW44JWA9ZhuiJ4SJvW72G1RD81Pp1WZk29)
ain a0 a4 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (¬ ain a1 a1) → (¬ (a0 = a2)) → ain a0 (asm (apow a2) (asing a2)) → ain a2 (asing a2) → (¬ ain a1 (apow (asing a2))) → (¬ (asing a1 = a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ (asing a1 = asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (¬ ain a3 (asing a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → ain a1 (aun a3 (asing (asing a2))) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 (apow (asing (asing a1))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal4 (aun a3 (asing (apow a2))) → aex2 (apow (asing a2)) → aex2 (aun (asing (asing a1)) (asing a3)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (asm a4 (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ ain (asing (asing a1)) a4)
In Proofgold the corresponding term root is 9bf5df... and proposition id is f5038f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVX6wZxmc7yyapsYmJKxjmoEugEDynFM5X)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → ain a1 a3 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a0 (asm a3 (asing a1)) → ain (asing a1) (apow a2) → (¬ ain (asing a1) a2) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (a1 = asm (apow a2) a2)) → (¬ (apow (asing a1) = a3)) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun a4 (asing (apow a2))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (apow a2) (asing (asing a2))) → aal5 (aun (asing (asing a1)) a4) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm a3 (asing a1)) → aex2 (asm a4 a2) → aex2 (aun a1 (asing (apow a2))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (asm a4 (asing a3)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq a2 (aun (asing a1) (apow (asing a2)))
In Proofgold the corresponding term root is bed426... and proposition id is c978c5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMPeugjTAHH1TZ68oGTckSFfNpUMgRjUksr)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → ain a0 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a2 a1) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (apow a2) → (¬ (a0 = asm (apow a2) a2)) → (¬ (a1 = asing a2)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain a1 (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asing a2)) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (asing a1) a3) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (¬ ain (asing a2) a3) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (asm (apow a2) a2 = apow a2)) → ain a0 (apow (asing (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asing a2) (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a2 (asing a1) = a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing a3)) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal2 (asm (apow a2) a2) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26)
In Proofgold the corresponding term root is b17304... and proposition id is 9058de...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMwnXJqAotfCKGkb9rMmMcspGqV3Jepgfj)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → asubq a1 a2 → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a1 (aun (asing a1) (asing (asing a1))) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (asing a1 = a2)) → ain a2 (asm (apow a2) a2) → (¬ ain a0 (asm (apow a2) a2)) → ain a2 (asm (apow a2) (asing a1)) → (¬ (a1 = asing a2)) → (¬ asubq (asing a1) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asm (apow a2) a2) a2) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (¬ asubq (apow a2) (apow (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a3 (asing a3) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → ain (asing a1) (aun (asing (asing a1)) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (asm (apow a2) a2) → aex2 (asm a4 a2) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun a3 (asing (apow a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun a4 (asing (apow a2))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25)))
In Proofgold the corresponding term root is 229904... and proposition id is d0693a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGDTJvWEGTHGLFFtNc4o4kWHJJA6pucvnG)
(∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → ain a2 a3 → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (¬ ain a3 a2) → ain a2 (asing a2) → ain (asing a1) (asm (apow a2) a2) → (¬ (a1 = asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing a2) a3) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a0 (apow (apow (asing a1))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain a0 (asing a3)) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (asm a2 (asing a1)) a1 → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a4 (asing a3)) a3 → asubq a3 (aun (apow a2) (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))
In Proofgold the corresponding term root is d806b7... and proposition id is 220535...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJgXHWVF6r42gFhkvZ26DjzDiTa8PdP5Ne)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → ain a0 a2 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ (a2 = apow a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (asm a3 (asing a1) = a3)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain a3 (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain (apow a2) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm a4 a2) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → asubq a2 (asm a4 (asing a2)) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (apow (asing a2))) (asing a2)) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (¬ aal2 (asing a3)) → (¬ aal5 a4) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal5 (aun (asing (asing a1)) a4) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (asm a4 a2) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (apow a2) (asing (apow a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25))
In Proofgold the corresponding term root is 1ad78a... and proposition id is 44bd64...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTV4EugiZqzLBHsjmW2SRS62diBSVifk2k)
(∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → asubq a2 a3 → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ (a0 = asing a1)) → (¬ ain a1 (asing a2)) → (¬ (a0 = asm (apow a2) a2)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq a2 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ (a1 = apow (asing a1))) → (¬ (asing a1 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = apow a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ (a3 = asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → ain (asing a2) (asing (asing a2)) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a1)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 a1) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 (asm a4 (asing a2)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25))
In Proofgold the corresponding term root is 5d124e... and proposition id is db936b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbLdkxFu9PXdmfTKKUmFafxM8hXWzPoLWD)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → ain a0 a1 → ain a3 a4 → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (¬ (a0 = a2)) → (¬ asubq a1 (asing a2)) → ain a2 (apow a2) → ain a2 (asm (apow a2) (apow (asing a2))) → (¬ ain a3 (asing a1)) → (¬ (a1 = asing (asing a1))) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (asing a1 = apow a2)) → (¬ ain (asing a1) (asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asing (asing a1)) a3) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain (asing a2) a4) → ain a0 (aun (asing a2) (apow (asing a2))) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a0 (asm a4 a2)) → ain a3 (asm a4 a2) → ain a3 (asm a4 (asing a1)) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → (¬ asubq (aun a4 (asing (apow a2))) a4) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm a4 (asing a3) = a3) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal2 (asm a4 a2) → aal5 (aun a4 (asing (apow a2))) → aex2 a2 → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex3 (asm a4 (asing a3)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2)))
In Proofgold the corresponding term root is c1b781... and proposition id is ec1d65...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbuMmpZFDHYAQTpYH6F2E6R652GJWfGRQM)
(∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (asing a1) a0) → ain a0 (apow (asing a1)) → ain a0 (asm a3 (asing a1)) → (¬ asubq a1 (asing a2)) → ain a2 (asm a3 (asing a1)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing (asing a1)) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → asubq a3 (apow a2) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a2 (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → asubq a4 (aun (asing (asing a1)) a4) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq a2 (asm a3 (asing a2)) → (asm (asm a3 (asing a1)) (asing a2) = a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal4 a3) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 a4 → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 a2 → aex4 (aun a3 (asing (asing a2))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24)
In Proofgold the corresponding term root is 95ef00... and proposition id is accdc1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQ2t5CfdU1Jr6pXgdhXDjusXzXBXWNmLMc)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → ain a0 a1 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ain X24 (apow (asing X25)) → ((X24 = a0) ∨ (X24 = asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain a1 (aun (asing a1) (asing (asing a1))) → ain (asing a1) (asm (apow a2) a2) → (¬ asubq (asing a1) (asing a2)) → (¬ ain a1 (asing (asing a1))) → (¬ (a1 = asing (asing a1))) → (¬ (asing a1 = a3)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a2)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm a2 (asing a1)) a1 → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (apow (asing a2)) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal3 a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (apow (asing a2))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal3 (aun (asing a2) (apow (asing a2))) → aal3 (aun a2 (asing (apow a2))) → aal4 a4 → aal5 (aun (asing (asing (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (apow a2) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3)))
In Proofgold the corresponding term root is 9c26d0... and proposition id is 9042d0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMH588FyWvVCMJ4usYFD53gxAUshdWE1eto)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a0 a3 → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → (aal3 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a0 (asm a3 (asing a1)) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain a2 (apow (asing a1))) → (¬ ain (asing a1) (apow (asing a2))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ ain (apow a2) (apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = apow a2)) → (¬ (asing a2 = asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain (asing (asing a1)) (apow (asing (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain (asing a2) (apow (asing a2)) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal3 (apow (asing a2))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal3 (aun (asing a2) (apow (asing a2))) → aal5 (aun a4 (asing (apow a2))) → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun (asing (asing a1)) a4) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (¬ ain a3 (asing (apow a2)))
In Proofgold the corresponding term root is c826ef... and proposition id is df44c9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdSFo3JLWqextQ6qtunre3aiU17NL32Hd9)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → asubq a3 a4 → (¬ ain a1 (apow (asing a1))) → (¬ ain a2 (apow (asing a2))) → (¬ asubq (asm (apow a2) a2) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = a3)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ ain a3 (apow a2)) → (¬ (a2 = apow a2)) → asubq (asm a3 (asing a1)) (apow a2) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a2) (asing a3)) → ain a0 (aun a1 (asing a3)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain a3 (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (asm a4 (asing a2)) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → (asm a4 (asing a3) = a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) a4 → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing a3)) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm a4 a2)) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 a4) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal2 (asm (apow a2) (apow (asing a1))) → aal2 (apow (asing (asing a1))) → aal3 a3 → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (aun (asing a3) (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asing (asing (asing a1))) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal3 (apow (asing (asing a1))))
In Proofgold the corresponding term root is c27ead... and proposition id is b3870e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcXtGjcyEEiY4RE4DSCuTzs2SqDcPA2q3R)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ (a0 = asm a3 (asing a1))) → (¬ (a1 = apow (asing a1))) → (¬ ain (asing a2) a2) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = apow a2)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (asing a2 = a3)) → (¬ (asm a3 (asing a1) = a3)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ (asing a2 = apow a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain a0 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal3 (apow (asing a2))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (aun a1 (asing a3)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (¬ (asing (asing a1) = apow (asing a1)))
In Proofgold the corresponding term root is 6f7d14... and proposition id is a4ab84...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbBYUFiBKojfgoWad2vSkJiWVLRgVptnx1)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, (aun X24 X25 = aun X25 X24)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ asubq a1 (asing a2)) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ (asing a1 = asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asm (apow a2) a2)) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → asubq (asm a3 (asing a1)) (apow a2) → (¬ (a3 = asm (apow a2) a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (apow a2) (asing (asing a2))) → ain a1 (apow (asm a3 (asing a1))) → ain a0 (aun a1 (asing a3)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asing a1) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aun (asing a2) (asing (asing a2))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1
In Proofgold the corresponding term root is af2da2... and proposition id is e0d1dc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJDs45bqgjhKh6vSrt2V5FxczDbyt3J3sT)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ (a0 = a1)) → ain (asing a1) (asing (asing a1)) → ain a1 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (apow (asing a1)) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a2 = asm (apow a2) a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (asm a3 (asing a1) = a3)) → (¬ (a3 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a0 (aun (asing a1) (apow (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a3 (asing a3) → ain a0 (asm a4 (asing a2)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (asm a2 (asing a1) = a1) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (asm (asm a4 a2) (asing a2) = asing a3) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (asm a4 (asing a3) = a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (asm a3 (asing a1)) → aal2 (apow (asing (asing a1))) → aal4 (aun a3 (asing (asing a2))) → aex2 (asm a3 (asing a1)) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0))
In Proofgold the corresponding term root is f08d95... and proposition id is 98f4e5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGdW9rwZFbvv6QqhPJNv74sDn7CANe3n5L)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (¬ (a1 = a2)) → (¬ asubq a2 a0) → ain a0 (apow a2) → (¬ (a0 = asm a3 (asing a1))) → ain (asing a1) (apow (asing a1)) → asubq a1 (apow (asing a1)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asing a2) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = a3)) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a3 (asm a4 (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (asing (apow a2)) → ain a0 (aun a1 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing a3)) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal4 (aun a3 (asing (asing a2))) → aex2 (asm a3 (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun a3 (asing (asing a2))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))
In Proofgold the corresponding term root is 1ed42c... and proposition id is bc3e74...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVeAcQAMwMjQjeRk6224cgS5a5WYZP6FEe)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a2 a2) → ain a1 (asm (apow a2) (apow (asing a1))) → (¬ (a1 = asing a1)) → ain a2 (asm (apow a2) (asing a1)) → ain a0 (apow (asing a2)) → (¬ ain (asing a2) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (asing (asing a1) = a3)) → (¬ (asing a2 = a3)) → asubq a3 (apow a2) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → ain a3 (asing a3) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (asing a2))) a3) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (asm a2 (asing a0) = asing a1) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq a3 (asm a4 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal2 (asing a3)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 (asm (apow a2) (asing a1))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 a3 → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aun (asing a3) (asing (apow a2))) → aex3 a3 → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex4 (aun a3 (asing (asing a2))) → aex3 (asm a4 (asing a3)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0)))
In Proofgold the corresponding term root is e9dad5... and proposition id is 18cf02...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbNiSf8u6GdKrJ8D3BkN3dgFwPVZin4wpw)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (¬ asubq a3 a2) → asubq (asing a1) a2 → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a2 (asing a1)) → ain a0 (apow (asing a2)) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ (a2 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain a3 (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) a4) → ain a1 (aun (asing a1) (asing a3)) → ain a2 (asm a4 (apow (asing a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (asm (asm a4 (asing a2)) (asing a1)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 (asing a2)) (asing a3) = a2) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a4 (asing a3)) a3 → asubq a3 (asm a4 (asing a3)) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal4 (aun a3 (asing (asing a2))) → aex2 (asm a3 (asing a1)) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2))) (asm a4 (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, (¬ aal2 (asing X24)))
In Proofgold the corresponding term root is b97dfa... and proposition id is fd8f63...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUhRdeP8UWevuTWMCqV6YDfMk12bV54EpT)
(∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (¬ ain a3 a2) → (¬ ain a3 a3) → (¬ asubq a3 a2) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ ain (asing a1) a2) → ain (asing a1) (asm (apow a2) a2) → (¬ (a0 = asing (asing a1))) → (¬ ain a1 (asing a2)) → (¬ ain (asing a2) a1) → (¬ ain a2 (asing a1)) → asubq a1 (asm a3 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing a2 = apow a2)) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (asm a3 (asing a1)) a4) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing a1)) → (¬ aal2 (asing (asing a1))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 a4 → aal5 (aun (asing (asing a1)) a4) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → ain (asing a1) (apow (asing a1))
In Proofgold the corresponding term root is 1459c5... and proposition id is 055e19...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJdJWy8dTUzZ1vterRDDdShT7JynQw4RSV)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → ain a2 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ asubq a2 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ asubq a1 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ asubq (asing a2) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (asing a1) a3) → (¬ (a2 = asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (asm a3 (asing a1) = a3)) → asubq a3 (apow a2) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a0 (aun a3 (asing (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asing (asing a1)) a4) → adisjoint (apow (asing a1)) (asm a4 a2) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal2 (asm (apow a2) (apow (asing a1))) → aal4 (aun a3 (asing (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm a3 (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (apow a2) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aex2 (asm (apow a2) (apow (asing a1)))
In Proofgold the corresponding term root is af0295... and proposition id is 098f24...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWmATEyFabRsd54HDsWqrMQpa9cjpdGgMY)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (¬ asubq a2 a0) → (¬ asubq a1 (asing a1)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → (¬ (a0 = asing a1)) → ain a2 (apow a2) → (¬ ain a2 (apow (asing a2))) → ain a2 (asm (apow a2) (apow (asing a2))) → ain a0 (apow (asing a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (asing a2) a3) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ (asm a3 (asing a1) = apow a2)) → ain a1 (apow (apow (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → ain a3 (asm a4 a2) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → (X24 = aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → aal2 (aun (asing a1) (asing (asing a1))) → aal4 (aun a3 (asing (asing a2))) → aex3 a3 → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25)))
In Proofgold the corresponding term root is da558f... and proposition id is eb6ddc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTc23DtwcBUtX9n77ytmCEzAVFzehh1ovf)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ ain a1 a0) → asubq a2 a3 → (¬ (a0 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (¬ ain a1 (asing a2)) → (¬ ain a2 (apow (asing a1))) → ain a2 (asm (apow a2) (asing a1)) → ain a0 (apow (asing a2)) → (¬ asubq (asing a1) (asing a2)) → (¬ (asing a1 = apow a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ (a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain a1 (asing (apow (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain a3 (aun a1 (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (asm a2 (asing a1) = a1) → asubq a2 (asm a3 (asing a2)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → (asm a4 (asing a3) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (¬ aal5 a4) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 (asm (apow a2) a2) → aal3 (asm (apow a2) (asing a1)) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal5 (aun (asing (asing a2)) a4) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (asm a4 a2) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (asm (apow a2) (asing a0)) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1))))
In Proofgold the corresponding term root is af262a... and proposition id is 694b33...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMadMbq3yPRshETXtJkYLzaGTzNkx7JxXtZ)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a0 a3 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (¬ ain a3 a3) → (¬ asubq a3 a2) → asubq a3 a4 → (¬ (a1 = a2)) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (¬ ain a0 (asing a2)) → ain a2 (asing a2) → ain a1 (asm (apow a2) (apow (asing a2))) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing a1) (asing a1)) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain a3 (asing a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → ain a0 (asm a4 (asing a1)) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 a3) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun a1 (asing a3)) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (asm a4 (asing a0)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq (asing a1) a2
In Proofgold the corresponding term root is b28971... and proposition id is d2502a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJrg2qSjbpFeFVbnjozf51Bgv83twLCEy4)
(∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a0 a0) → (¬ ain a2 a0) → asubq a1 a2 → asubq a3 a4 → ain (asing a1) (asing (asing a1)) → ain (asing a1) (apow a2) → (¬ asubq a1 (asing a2)) → (¬ (a1 = asing (asing a1))) → (¬ ain (apow a2) a2) → (¬ ain a2 (asing (asing a1))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = apow a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain (asing (asing a1)) (asing (asing (asing a1))) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a4 (asing (asm (apow a2) a2))) → (¬ ain a3 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (aun (asing a2) (apow (asing a2))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (asing a0)) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 (asing (asing a1))) → (¬ aal2 (asing a2)) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow a2) (apow (asing a2))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))
In Proofgold the corresponding term root is a35faf... and proposition id is 68a401...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdGmc8hu3b9pLcNajdNrVendjWfpx3nUTQ)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal3 (aun (asing X24) (asing X25)))) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a0 (apow (asing a1)) → (¬ (a0 = asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a1 = apow (asing a1))) → (¬ asubq (asing a2) a2) → (¬ asubq a2 (asm a3 (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) a3) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ ain (asing (asing a1)) a3) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain a0 (apow (asing (asing a1))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain (asing (asing a1)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun a4 (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a3 (asing (apow a2))) a3) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → asubq (asing a1) (asm (asm (apow a2) (apow (asing a1))) (asing a2)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (asm (apow a2) (asing a1)) (asing a2)) (apow (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal3 (asm a4 a2)) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asing (asing a1)) a4) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (asm (apow a2) a2) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2)))
In Proofgold the corresponding term root is bfb08a... and proposition id is c03abc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUbfv2JydCNiFpqUdJ4pPfjSGuU3MzLip7)
ain a0 a1 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X25 X26 → asubq (aun X24 X25) (aun X24 X26)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → asubq (asing a1) a2 → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ asubq a2 (asing a1)) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ asubq (asing a2) a2) → (¬ ain a2 (asing (asing a1))) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = apow a2)) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) a3) → (¬ (asing (asing a1) = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a2 (aun a3 (asing (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a0 (aun a3 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1)) = apow (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq (asm a3 (asing a1)) a4 → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → aal3 a3 → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (asm a4 (asing a2)) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1))
In Proofgold the corresponding term root is d0f7d9... and proposition id is 3c5799...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYSjSAmGiU93m6Mu5ebXkC6KPhsHURKP7f)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (¬ ain a2 (apow (asing a2))) → (¬ asubq (asing a1) (asing (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain (asing a2) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) a2) a2) → asubq (asing (asing a1)) (apow (asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a3 (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → aal3 (asm a4 (asing a1)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex5 (aun a4 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a1)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → ain (asing a2) (aun (asing a2) (apow (asing a2)))
In Proofgold the corresponding term root is b09fa0... and proposition id is fd7290...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZ8VVUuueeS1JWjsS5krtHKkeMozNoMrRB)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a1 a2 → ain a0 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a2 a2) → asubq a1 a2 → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq a1 (asing a1)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a2 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a2))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (a1 = apow (asing a1))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow (asing a1)) a3) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain a0 (apow (asing (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain a3 (asm a4 (asing a2)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a1 (aun a2 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → (¬ asubq (aun a3 (asing (apow a2))) a3) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 a1) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (aun (asing a1) (apow (asing a2))) → aal3 (aun (asing a2) (apow (asing a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 a2 → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (apow a2) → aex3 (asm (apow a2) (asing a0)) → aex5 (aun (asing (apow (asing a1))) a4) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)))
In Proofgold the corresponding term root is 67ae45... and proposition id is 819b96...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNgPFLP7cB5K45qu8SrmfRwmWmYUCR1Vmy)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → ain a2 a3 → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain (asing a1) (apow a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (asing a1 = a2)) → (¬ (a2 = asing (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → ain a0 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain a2 (aun a1 (asing a3))) → ain a3 (aun a1 (asing a3)) → (¬ ain (asing (asing a1)) a4) → (¬ ain (apow a2) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a2) (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing a2) (apow (asing a2))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aun a3 (asing (apow a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))
In Proofgold the corresponding term root is 700d6d... and proposition id is b6c6ad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXmZm59zjDXrEMtJS3XGSEtNAKc9CxiprQ)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ (asing a1 = a2)) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq a3 (asing a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (asing (asing a1) = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → ain (asing a2) (asing (asing a2)) → ain a1 (aun a2 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asing a2) (asm (asm a4 a2) (asing a3)) → (asm (asm a4 a2) (asing a3) = asing a2) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq (asm a4 (asing a3)) a3 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 (aun a1 (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex3 (asm a4 (asing a1)) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → ain a2 (asm a3 (asing a1))
In Proofgold the corresponding term root is 23f5b0... and proposition id is cdd768...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZFQcusAgjv7fATw2dM7f92iPQWTD6VWe7)
ain a0 a4 → ain a1 a4 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a2 a2) → (¬ (a0 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ (a0 = asing a2)) → (¬ (a0 = asm a3 (asing a1))) → ain (asing a1) (apow (asing a1)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ asubq a2 (asing a2)) → (¬ ain a1 (asing (asing a1))) → (¬ ain a2 (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (¬ (a2 = asing a2)) → (¬ asubq a3 (asing a2)) → (¬ ain (asing a2) (asm a3 (asing a1))) → (¬ (a1 = apow a2)) → (¬ (asm (apow a2) a2 = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain a0 (apow (asing (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asing a2) a4) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a1 (asm a4 (asing a2)) → (¬ ain a3 (aun (apow a2) (asing (apow a2)))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1) = aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 a2 → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (asm a4 (asing a0)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))))
In Proofgold the corresponding term root is 1f4b85... and proposition id is 44c31b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTNksrZ4UQGLJVUfBEgUxfkdcvZfxezJco)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a0 a3 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → asubq a1 a2 → asubq a3 a4 → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a2 (apow a2) → (¬ ain a1 (asm (apow a2) (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq a3 (apow a2) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ (a3 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain a1 (apow (apow (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain (asing a2) (asing a3)) → ain a3 (asing a3) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (¬ asubq (asm a4 (asing a2)) a2) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (apow (asing a1))) a4) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal2 (asing (apow a2))) → (¬ aal3 (apow (asing a1))) → (¬ aal5 (aun a3 (asing (asing a2)))) → aal2 (apow (asing (asing a1))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 076696... and proposition id is 245e9a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZnZCzUMgG4kFDtduRLt4zUAsMedxNApEf)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal5 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal4 X25))))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → ain a0 a1 → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal2 X24 → aal2 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a1 a0) → (¬ asubq a3 a2) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a1 (apow a2) → (¬ ain (asing a1) a2) → ain (asing a1) (apow (asing a1)) → (¬ ain (apow a2) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ (a2 = apow (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ ain (apow a2) (apow a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (asing (asing a1) = a3)) → (¬ (asing a2 = apow a2)) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain a3 (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a1 (asing a3)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a0 (asing (apow a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → asubq (asm (apow a2) (asing (asing a1))) a3 → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) a4 → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 a2 → aal3 a3 → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is 80d9c8... and proposition id is dd4ddd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMaBexEUkyeGx7xEFHiCNCfPP35G3QDsEw)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (ain X24 a2 ↔ ((X24 = a0) ∨ (X24 = a1)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → ain a0 a4 → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (¬ ain a1 a0) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → ain (asing a1) (asm (apow a2) (asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ (a1 = asing (asing a1))) → (¬ ain (asing a2) a2) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ asubq a3 (asing a2)) → (¬ ain (apow (asing a1)) a3) → (¬ (a0 = apow a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain a1 (asing a3)) → ain a1 (asm a4 (asing a2)) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → asubq a2 (asm a4 (asing a2)) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = asing a1)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing a1)) → (¬ aal3 (asm a4 a2)) → (¬ aal4 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2))))) → (¬ aal5 a4) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun a4 (asing (apow a2)))) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal4 (apow a2) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex2 a2 → aex2 (aun (asing (asing a1)) (asing a3)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25))))))
In Proofgold the corresponding term root is ba9370... and proposition id is b37001...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdHbJqEWQpU49Y2KUMVZiNBMzTGaCcdDWJ)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ asubq a1 (asing a2)) → (¬ ain a1 (apow (asing a2))) → (¬ (a0 = apow (asing a1))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (apow a2) a2) → (¬ (asing a1 = a3)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a1 = apow a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ (apow (asing a1) = a3)) → (¬ (asing a2 = a3)) → (¬ ain (asing a2) (asm (apow a2) a2)) → ain a0 (aun a3 (asing (asing a2))) → ain a1 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → ain a1 (asm a4 (asing a2)) → (¬ ain (apow (asing a1)) a4) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain a0 (asm a4 a2)) → ain (apow a2) (asing (apow a2)) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a4 (aun a4 (asing (apow a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a2 (asm a3 (asing a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal3 (asm (apow a2) (apow (asing a2))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal5 (aun a4 (asing (apow a2))) → aex2 (apow (asing a1)) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24)
In Proofgold the corresponding term root is 41c9b8... and proposition id is 6994b3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYMYZS2iytpHggiwvbeZpoEczMLbJuFGz4)
(∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (a1 = asing a0) → (¬ asubq a3 a2) → (¬ (a1 = a2)) → (¬ ain (asing (asing a1)) a2) → (¬ asubq (asing a2) a2) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = a3)) → (¬ (asing a1 = asm (apow a2) a2)) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → adisjoint (apow (asing a1)) (asm a4 a2) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun a4 (asing (apow a2))) a4) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (apow a2) (asing a0) = asm (apow a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → aal2 (apow (asing (asing a1))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex2 (apow (asing a2)) → aex2 (asm a4 a2) → aex3 a3 → aex3 (aun (asing a2) (apow (asing a2))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun (asm (apow a2) a2) (apow (asing a2))) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))
In Proofgold the corresponding term root is f8dcf8... and proposition id is 51051b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMQRLJ7sYx2PimP8CcXQ32VTiERYvtQnshR)
(∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ (a1 = a3)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (¬ asubq a2 (asing a1)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (¬ asubq (apow a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (apow a2) (apow a2)) → (¬ ain (apow a2) a3) → asubq (asm (apow a2) (apow (asing a1))) (asm (apow a2) (apow (asing a2))) → (¬ (asing a2 = apow a2)) → ain a0 (apow (asing (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → (¬ ain (apow a2) (asing (asing a2))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (asing a1) a4) → ain a2 (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a2 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → ain a3 (asm a4 (asing a1)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a1 (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm a4 (asing a0)) (asm a4 (apow (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal3 (asm (apow a2) (asing a2)) → aex2 (apow (asing a1)) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ (a3 = asm (apow a2) a2))
In Proofgold the corresponding term root is f35761... and proposition id is 2029ea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWPPBfBYaYBhEdyn4K7adicow5YY8Bnwus)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (¬ asubq a4 a3) → (¬ asubq a2 a0) → (¬ ain (asing a1) a0) → (¬ ain (asing a1) a1) → (¬ ain a1 (asing a2)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ asubq (asing a2) a2) → (¬ asubq a2 (asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ (a2 = asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain a0 (asing a3)) → ain a2 (asm a4 a2) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain (asing a1) (asing (apow a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → asubq a2 (asm a4 (asing a2)) → asubq (aun a3 (asing (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a2) (apow (asing a2))) (asm a3 (asing a1))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing a2)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex2 (asm a3 (asing a2)) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))
In Proofgold the corresponding term root is 31ffd3... and proposition id is 6dbfe0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMS8TgJvo8yxTN9VVVnpk8wSTratX1RM7SM)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a2 (asing a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → ain (asing a1) (asm (apow a2) (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → asubq (asm a3 (asing a1)) (asm (apow a2) (asing a1)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → ain a1 (apow (apow (asing a1))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain a3 (asm a4 a2) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a1 (aun a2 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (asm a3 (asing a2) = a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm a4 (asing a1)) → aal4 a4 → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (asm a4 (asing a1)) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25))
In Proofgold the corresponding term root is 92d630... and proposition id is 13f2a4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMRiJpkmja3iMN5iw15ga38d3akrwd53aRa)
(∀X24 : set, (¬ ain X24 X24)) → ain a2 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → asubq a3 a4 → (¬ asubq a4 a3) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (¬ ain (asing a1) a0) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (asing a1 = apow a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (asing a2 = apow a2)) → (¬ (a3 = apow a2)) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain (asm a3 (asing a1)) (aun a3 (asing (asm a3 (asing a1)))) → ain a1 (aun (asing a1) (asing a3)) → ain a3 (asm a4 (asing a1)) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → ain a0 (aun (asing (asing a2)) a4) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a0)) → ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a3)) → ain X24 a2) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 a2) (asing a2) = asing a3) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 a1) → (¬ aal2 (asing a1)) → (¬ aal2 (asing a2)) → (¬ aal4 (asm a4 (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aex2 (asm a4 a2) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun a3 (asing (apow a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1)) (aun (apow (asing a2)) (asing a3)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1))
In Proofgold the corresponding term root is 27b604... and proposition id is 574b54...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXhYqYQ5GcJ5cjdGyAxx82VGmeczpck5Wp)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a1 a3 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (¬ ain a0 a0) → (¬ ain a1 (asing a2)) → ain a1 (asm (apow a2) (asing a2)) → asubq a1 (apow (asing a1)) → ain a2 (asm (apow a2) (asing a1)) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ ain (apow a2) (asing (asing a1))) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asing a1) a3) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ ain a3 (apow a2)) → (¬ (a2 = asing a2)) → (¬ ain (asing (asing a1)) a3) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (asing a2 = a3)) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (asing (asing a2)) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (asing a1) a4) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → (¬ ain a0 (asing a3)) → (¬ ain a2 (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm a2 (asing a0)) (asing a1) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → aal5 (aun (asing (asing (asing a1))) a4) → aex2 (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex2 (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → aex3 (asm (apow a2) (asing (asing a1))) → aex5 (aun (apow a2) (asing (apow a2))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))))
In Proofgold the corresponding term root is ed861f... and proposition id is edc346...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWBCpmt2MB51BBEwwJK9QeiptUguXLkUbc)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → ain a0 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (¬ asubq a3 a2) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a0 (asm (apow a2) (asing a2)) → (¬ (a0 = apow (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a2) a2) → (¬ asubq a2 (asm a3 (asing a1))) → (¬ (asing a1 = asm a3 (asing a1))) → (¬ (asing a1 = a3)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain a3 (apow a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (apow (asing a1)) a3) → (¬ (asm a3 (asing a1) = a3)) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain a0 (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a2 (asm a4 (asing a1)) → ain a0 (aun (asing a2) (apow (asing a2))) → ain a0 (aun a3 (asing (asing a2))) → adisjoint (apow (asing a1)) (asm a4 a2) → ain a2 (asm a4 a2) → ain a2 (asm a4 (apow (asing a2))) → ain a0 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm a2 (asing a0)) (asing a1) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a0)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex3 a3 → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun (asing (asing a2)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → aex5 (aun a4 (asing (apow a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → ain a2 (asm (apow a2) (asing a1))
In Proofgold the corresponding term root is 5375b8... and proposition id is 241ab6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMa92XVesNQeVutWY78vY9c44SGXcmb84xX)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a2 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (¬ (a1 = a3)) → (¬ (a2 = a3)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a0 (asm a3 (asing a1)) → ain a1 (apow a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a1 (asing a2)) → (¬ (a0 = asing a2)) → (¬ (a0 = asm (apow a2) a2)) → (¬ asubq a2 (asing a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing a1 = asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (apow a2) (apow a2)) → (¬ ain (asm (apow a2) a2) a3) → asubq a3 (apow a2) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain a0 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) a4) → ain a0 (apow (asm a3 (asing a1))) → ain a0 (asm a4 (asing a2)) → ain a1 (asm a4 (asing a2)) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun a4 (asing (apow a2))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a3 (asing a2) = a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asing (asing a1)) (asm (asm (apow a2) a2) (asing a2)) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq a2 (apow (asm a3 (asing a1))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing (apow a2))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal4 (asm (apow a2) (asing a2))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal3 a3 → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asing (asing a2)) a4) → aal5 (aun a4 (asing (apow a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aex3 (asm a4 (asing a1)) → aex4 (aun a3 (asing (asing a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2)))
In Proofgold the corresponding term root is 9f84e0... and proposition id is 41ada3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMV1rzxq7EwipNEd3c5gYE82jhA4TWcYWTU)
(∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (¬ asubq a2 a1) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain (asing a1) (asing (asing a1)) → (¬ (a0 = asing a1)) → (¬ (a1 = asing a1)) → (¬ (a0 = asing (asing a1))) → (¬ ain a1 (asing a2)) → (¬ (a0 = asm (apow a2) a2)) → (¬ ain a0 (asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ (asing (asing a1) = a3)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a0 (aun a1 (asing a3)) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → ain a1 (aun (asing a1) (asing a3)) → ain a0 (asm a4 (asing a2)) → ain a1 (asm a4 (apow (asing a2))) → ain a3 (aun (asing (asing a2)) a4) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a1 (asm a2 (asing a1)) → asubq (asm a3 (asing a2)) a2 → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (apow (asing a2)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (¬ aal3 (asm (asm a4 (asing a2)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aal2 a2 → aal3 (aun a2 (asing (apow a2))) → aal4 (aun a3 (asing (apow (asing a1)))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a1 (asm a3 (asing a1)))
In Proofgold the corresponding term root is 169fb8... and proposition id is ede17a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMoUpBkpuN3fBBYVdMRbKbEcTffJbqeRLJ)
ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (¬ asubq a4 a3) → (¬ asubq a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (asing a1) a2) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a3 (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asing a2)) → asubq (asing (asing a1)) (apow (asing a1)) → (¬ ain (asing a1) a3) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain (asing (asing a1)) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → (¬ ain (asing a1) (asm a4 a2)) → ain a2 (asm a4 a2) → ain a3 (asm a4 (asing a1)) → ain a0 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq a1 (asm a2 (asing a1)) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal3 a2) → (¬ aal3 (apow (asing (asing a1)))) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → aal3 a3 → aal4 (aun a3 (asing (apow (asing a1)))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (apow a2))
In Proofgold the corresponding term root is 69917b... and proposition id is ab6738...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMUsuHRdKvH25D8fRMBcjrx9jAUoe7hmo2U)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (¬ ain a1 a0) → (¬ (a1 = a2)) → ain a0 (asm (apow a2) (asing a2)) → ain a2 (asing a2) → ain (asing a1) (apow a2) → ain (asing a1) (apow (asing a1)) → (¬ ain a2 (apow (asing a2))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ (asing (asing a1) = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a2 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain (asing a1) X24 → ain a1 X24 → asubq (aun (asing a1) (asing (asing a1))) X24) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (asm (apow a2) a2) (asm (asm (apow a2) (apow (asing a2))) (asing a1)) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → asubq (asm a3 (asing a1)) (aun (asing (asing a1)) a4) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (aun a1 (asing a3))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 (apow (asing (asing a1))) → aal3 (aun a2 (asing (apow a2))) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (asm a4 (asing a0)) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex5 (aun (asing (apow (asing a1))) a4) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1))))
In Proofgold the corresponding term root is 249e80... and proposition id is 155600...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFQnxxWTBS57bS1KSzC83hEG96eNePwQ1x)
(∀X24 : set, asubq X24 X24) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (asm X24 X25) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → ain (asing a1) (asm (apow a2) a2) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm (apow a2) a2) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a1 = apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (apow (asing (asing a1)))) → (¬ ain a0 (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain a0 (asm a4 a2)) → ain a1 (aun (asing (asing a2)) a4) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a2 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a4 (aun (asing (apow (asing a1))) a4) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a2 (apow (asm a3 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → (¬ aal2 (asing a2)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal4 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1)))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (asm a4 (asing a2)) → aal4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex2 (apow (asing a1)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun a3 (asing (asing a2))) → aex5 (aun (asing (asing (asing a1))) a4) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25))
In Proofgold the corresponding term root is 6fa575... and proposition id is 350511...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMJrmQBj31PzT7Z3srCKHMRtmaqR2avNXLU)
ain a2 a3 → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ (a0 = a3)) → (¬ (a1 = a3)) → (¬ ain (asing a1) a1) → (¬ ain (asing a2) a1) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ ain (apow (asing a1)) a3) → (¬ (a3 = asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → (¬ ain a2 (asing a3)) → ain a0 (asm a4 (asing a2)) → ain a1 (asm a4 (asing a2)) → (¬ ain (asing (asing a1)) a4) → ain a2 (aun (asing (asing a2)) a4) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1)) = aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal4 (apow a2) → aal5 (aun (asing (asing a1)) a4) → aex2 (apow (asing a1)) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (asm a4 (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex5 (aun (apow a2) (asing (asing a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ ain a3 (asing (apow a2)))
In Proofgold the corresponding term root is d8c2e4... and proposition id is 2a52cf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLdQWqTRQCVVbt3NWKBNQBs1hAPwtZ45Wg)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X24 → asubq X26 X25 → asubq X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aal2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal3 X24 → aal3 X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → asubq a3 a4 → ain a1 (asing a1) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (asing a1)) → (¬ asubq a1 (asing a1)) → (¬ ain a1 (apow (asing a1))) → (¬ ain a1 (asing a2)) → (¬ ain a2 (asing a1)) → asubq a1 (asm a3 (asing a1)) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ asubq a2 (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = apow a2)) → (¬ (a3 = asm (apow a2) a2)) → (¬ (a3 = apow a2)) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a0 (asing a3)) → (¬ ain (asing (asing a1)) a4) → ain a3 (aun (asing (asing a2)) a4) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing a1) (apow (asing a2))) a2) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a4 (aun (asing (asing a1)) a4) → asubq a4 (aun a4 (asing (asm (apow a2) a2))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (asing (asing a1))) a3 → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm (asm a4 (asing a1)) (asing a0)) (asm a4 a2) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → (asm a4 (asing a3) = a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 a1) → (¬ aal2 (asing a1)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aex2 (asm a3 (asing a1)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex5 (aun (asing (asing a2)) a4) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1)))
In Proofgold the corresponding term root is c16a40... and proposition id is ebac7d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNugAbVuTyqd9YP1JGLtcJFCebCvqqEJr3)
(∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → asubq a2 a3 → (¬ asubq a2 a0) → ain a1 (asing a1) → ain a0 (apow a2) → (¬ asubq a1 (asing a2)) → (¬ ain a2 (apow (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ asubq (asing a1) (asing a2)) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ ain a2 (asing (asing a1))) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ ain (asing (asing a1)) a3) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) a4) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → ain a1 (apow (asm a3 (asing a1))) → ain a2 (asm a4 a2) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a3 (aun a4 (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (apow (asing a1))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (asm (asm (apow a2) (apow (asing a1))) (asing a1) = asing a2) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a2)) (asing a1) = aun a1 (asing a3)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → asubq a3 (asm a4 (asing a3)) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (asm a3 (asing a1))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal5 (aun (apow a2) (asing (apow a2))) → aex2 (aun a1 (asing a3)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex4 (apow a2) → aex3 (asm (aun a3 (asing (asing a2))) (asing a2)) → aex4 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2)))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → asubq (asing (asing a1)) (asm (apow a2) (asing a2))
In Proofgold the corresponding term root is e489db... and proposition id is 0642d8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYemeqJYLZjTsDfaS7aGLnbU5VGNELjNFs)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → ain a1 a2 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ain a0 (apow X24)) → (∀X24 : set, ∀X25 : set, (¬ (X25 = X24)) → (¬ ain X25 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (¬ (X24 = X25)) → aex2 (aun (asing X24) (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (¬ ain a1 a1) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a0 (asm (apow a2) (asing a2)) → (¬ ain a1 (apow (asing a1))) → (∀X24 : set, ain X24 (asing a1) → (X24 = a1)) → asubq a1 (asm a3 (asing a1)) → (¬ ain a2 (apow (asing a2))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain a1 (asm a3 (asing a1))) → (¬ ain (asing a2) a2) → (¬ ain (apow a2) a2) → (¬ (asing a1 = a3)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (asing a2 = asm a3 (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ ain (apow a2) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → ain a0 (apow (asm a3 (asing a1))) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain (asing a2) (asing a3)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → (¬ ain a0 (asing (apow a2))) → (¬ ain (asing a2) (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (asm a2 (asing a0) = asing a1) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) (apow (asing a1))) (asing a2) = asing a1) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → asubq (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (asm a4 a2) (asing a3) = asing a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal7 (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aal4 (apow a2) → aal4 (aun a3 (asing (apow (asing a1)))) → aex2 (aun (asing (asing a1)) (asing a3)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a0) (asing (asm a3 (asing a1))))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex3 (asm (apow a2) (asing a0)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex5 (aun (asing (asing a2)) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → ain a2 (apow a2)
In Proofgold the corresponding term root is 0765df... and proposition id is 7d3030...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMWJvB1a5q6DSVScqg9HV3iTebiL4xe4fEQ)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (¬ asubq a4 a3) → (¬ asubq a1 a0) → ain (asing a1) (apow (asing a1)) → asubq a1 (asm a3 (asing a1)) → (¬ ain a2 (apow (asing a1))) → (¬ ain (apow a2) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (¬ (asing (asing a1) = apow (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm a3 (asing a1)) (asing a2)) → (¬ ain (apow a2) (asing a2)) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain a0 (asm a4 a2)) → ain a3 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a1 (aun (asing a3) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → asubq (aun (asing a1) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → aal2 a2 → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (apow (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (apow a2) → aex5 (aun (apow a2) (asing (asing a2))) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1))
In Proofgold the corresponding term root is cd68f0... and proposition id is 834b11...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcgi6nwbbcFD9eDmACgcKFeywWqVLS5YLd)
(∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → ain a3 a4 → (∀X24 : set, ain X24 a4 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ∀X25 : set, asubq (aun X24 X25) (aun X25 X24)) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a1 a1) → (¬ ain a3 a3) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (asing a1) → (¬ (a0 = apow (asing a1))) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain (asing a1) (asing a2)) → (¬ (a1 = asm a3 (asing a1))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ (asing a1 = asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a0 (asm a4 (asing a2)) → (¬ ain a2 (aun (apow (asing a2)) (asing a3))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ asubq (aun (apow a2) (asing (asing a2))) (apow a2)) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (¬ aal2 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal5 (aun (apow a2) (asing (asing a2))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a3)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (asing a1) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2)))) → (¬ ain a1 (asing (apow (asing a1))))
In Proofgold the corresponding term root is 933370... and proposition id is 8a7500...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcE4YD1LuAZXeVbtRz2dfJ82VS8Fnzbm2b)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (¬ asubq a1 a0) → ain a1 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (asm (apow a2) a2) → (¬ (a0 = asing (asing a1))) → ain a1 (asm (apow a2) (asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (apow a2) a2) → (¬ ain (apow a2) (asing (asing a1))) → (¬ (a2 = asm (apow a2) a2)) → (¬ ain a3 (asm a3 (asing a1))) → asubq a3 (apow a2) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (apow (asm a3 (asing a1))) → ain a3 (asm a4 (asing a2)) → ain a2 (asm a4 a2) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun a4 (asing (apow a2))) → asubq a3 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing a3))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (¬ (X24 = a1)) → ain X24 (aun a1 (asing a3))) → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a1) (apow (asing a2))) (aun (aun (asing a1) (asing (asing a1))) (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (apow a2)) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal2 (apow (asing (asing a1))) → aal3 (aun a2 (asing (apow a2))) → aal4 a4 → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun a4 (asing (apow a2))) → aex2 (asm (apow a2) a2) → aex5 (aun (asing (asing a1)) a4) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))
In Proofgold the corresponding term root is 983be2... and proposition id is d5c052...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMErG5qpL9DGgRH7Bjio8h1Df7moS2RjJRK)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → ain a0 a4 → (∀X24 : set, ∀X25 : set, ain X25 (asing X24) → (X25 = X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, asubq (aun (asm X24 X25) (aint X24 X25)) X24) → (∀X24 : set, aex2 (apow (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal6 X24 → aal5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a1 a0) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ ain a1 (apow (asing a1))) → asubq (asing a1) a2 → (¬ (a0 = apow (asing a1))) → (¬ ain a1 (asing a2)) → (¬ (a1 = asing a2)) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ (asing a1 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ (a1 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a2 (asm a4 (apow (asing a2))) → ain a2 (aun a3 (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a0 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) a2)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq (asm (apow a2) (asing (asing a1))) a3 → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (aun (asm (apow a2) a2) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (aun a3 (asing (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (apow (asing a2)) (aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (asm a4 (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → aal3 (asm (apow a2) (apow (asing a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (apow (asing a1)) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2)))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a2)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a3)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a3)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2))
In Proofgold the corresponding term root is 1a5733... and proposition id is aefcc6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcKFV4385dqJBxbgPVZGCe3xA6vF9uT7UP)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → ain a1 a3 → ain a2 a3 → ain a2 a4 → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → ain a0 (asm a3 (asing a1)) → asubq (asing a1) a2 → (¬ ain (asing a1) a2) → ain a2 (asm (apow a2) (asing a1)) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain (asing a1) (asing a1)) → (¬ asubq (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain a3 (asing a2)) → asubq (asing a2) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = asm (apow a2) a2)) → (¬ (a1 = apow a2)) → (¬ ain (apow a2) a3) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a1 (aun (asing a1) (asing a3)) → ain a3 (aun (asing a1) (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a2 (asm a4 a2) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a0 (aun a2 (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a0 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = a4) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal3 (apow (asing a2))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (aun a2 (asing (apow a2)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → aal2 (asm (apow a2) a2) → aal2 (apow (asing (asing a1))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → ain a0 (apow (aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is 782722... and proposition id is e4cbb3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMdsnmrj9nmTqESUmyJsN4mNZLFkyor9FhB)
(∀X24 : set, (aal3 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal2 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, adisjoint (asm X24 X25) (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → asubq a1 a2 → (¬ asubq a2 a1) → (¬ (a1 = a3)) → (¬ asubq a1 a0) → (¬ ain a1 (apow (asing a2))) → (¬ ain a0 (asing (asing a1))) → (¬ ain a2 (asing a1)) → (¬ ain a0 (asm (apow a2) a2)) → ain a0 (apow (asing a2)) → (¬ (asing a1 = apow a2)) → (¬ (a2 = asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ (apow (asing a1) = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (asing (asing a2))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asing a2) (asing a3)) → ain a3 (asm a4 (asing a1)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal2 (asm (apow a2) a2) → aal3 (asm a4 (asing a2)) → aal5 (aun a4 (asing (apow a2))) → aex3 (asm (apow a2) (asing a2)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex3 (asm a4 (asing a0)) → aex3 (asm a4 (asing a3)) → aex4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ aal5 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → ain (asing a2) (aun (asing a1) (apow (asing a2)))
In Proofgold the corresponding term root is 55de1c... and proposition id is 638e88...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNZUe8KdPmx9NU4K7zy5tRpsKybM37SML9)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun X24 (aun X25 X26)) (aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → ain a0 (asm a3 (asing a1)) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain a0 (asing (asing a1))) → ain a2 (asm (apow a2) (apow (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asing a2) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ asubq (apow a2) (asing (asing a1))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ (a2 = asing a2)) → (¬ (a1 = apow a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ (asing a2 = apow a2)) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a1 (apow (apow (asing a1))) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → (¬ ain (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (asing a2) (asing a3)) → ain a0 (aun a1 (asing a3)) → ain (asing a2) (aun (asing (asing a2)) a4) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a2 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a4 (aun (asing (asing a2)) a4) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2))) → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → asubq (asm a3 (asing a1)) (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal4 (asm a4 (apow (asing a2)))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal3 (aun (asing a1) (apow (asing a2))) → aal4 a4 → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asing (asing (asing a1))) a4) → aal5 (aun (asing (asing a2)) a4) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex3 (asm (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2))
In Proofgold the corresponding term root is 0cf174... and proposition id is fc8bb3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMMDGNwe93rtybyjX4iT1QmJp6ABbWGvtQr)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, (¬ ain X24 a0)) → ain a3 a4 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → aal2 (aint X24 X25) → (¬ aal4 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (¬ asubq X26 X24) → (¬ asubq X26 X25) → (¬ asubq (aun X24 X25) X26) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (∀X24 : set, ∀X25 : set, asubq X25 (asing X24) → ((X25 = a0) ∨ (X25 = asing X24))) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) a2) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain a1 (asing (asing a1))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm (apow a2) a2)) → asubq (asing (asing a1)) (asm (apow a2) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (apow a2)) → (¬ ain (apow a2) (apow a2)) → (¬ ain (apow a2) a3) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (asing (asing a2))) → ain a2 (asm a4 (asing a1)) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain (apow (asing a1)) a4) → (¬ ain (asm a3 (asing a1)) a4) → ain a1 (asm a4 (apow (asing a2))) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → (∀X24 : set, ain X24 (asm a4 (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun (asing (asing (asing a1))) a4) → asubq a2 (asm a3 (asing a2)) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 a2) (asing a3)) (asing a2) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a3)) → ain X24 a3) → asubq (asm a4 (asing a3)) a3 → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal4 (asm a4 (asing a2))) → (¬ aal4 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a2))) → (¬ aal6 (aun a4 (asing (asm (apow a2) a2)))) → aal4 (aun a3 (asing (apow a2))) → aal5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex2 (asm a4 a2) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm a4 (asing a0)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24)
In Proofgold the corresponding term root is fa2a64... and proposition id is 03b0f7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVApqodp2oY6kVih72QbrsLxYoa39cpj5W)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (¬ ain X24 X24)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, (¬ aal3 (apow (asing X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (¬ ain a3 a2) → (∀X24 : set, asubq X24 (asing (asing a1)) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → ain a1 (apow a2) → (¬ ain a0 (asing (asing a1))) → (¬ (asing a1 = a2)) → ain (asing a1) (asm (apow a2) (asing a2)) → (¬ asubq (asing a2) a2) → (¬ (asing a1 = asm (apow a2) a2)) → (¬ ain (apow a2) (asing (asing a1))) → (¬ ain (asing a1) a3) → (¬ ain (asing a2) (apow a2)) → (¬ asubq a3 (asing a2)) → (¬ ain (apow (asing a1)) a3) → ain a0 (apow (apow (asing a1))) → (¬ ain a0 (asing (apow (asing a1)))) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing (asing a1)) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing a2) (aun (apow a2) (asing (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain (apow (asing a1)) a4) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a2 (asm a4 a2) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (¬ asubq (aun (asing (asing (asing a1))) a4) a4) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) (apow (asing (asing a1))) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a3)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal2 (asing a1)) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (¬ aal5 a4) → (¬ aal5 (aun a3 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal6 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1))))) → aal3 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → aal3 (asm a4 (asing a1)) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aun a2 (asing (apow a2))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 a4 → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a2)) a4))
In Proofgold the corresponding term root is 0a103a... and proposition id is a9db2d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMZU9J167NhsPFwFzrr7vtDFPRcLVYqjvxj)
(∀X24 : set, (aal2 X24 ↔ (∃X25 : set, (ain X25 X24 ∧ (¬ asubq X24 (apow X25)))))) → (∀X24 : set, ain X24 a2 → ((X24 = a0) ∨ (X24 = a1))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a2 a0) → asubq a3 a4 → (¬ asubq a4 a3) → ain a1 (asing a1) → ain a0 (asm (apow a2) (asing a2)) → ain (asing a1) (asm (apow a2) a2) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain (asing a2) a1) → ain (asing a1) (asm (apow a2) (asing a2)) → ain a2 (asm (apow a2) (apow (asing a1))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ (a1 = apow (asing a1))) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ ain (asing a2) (apow a2)) → (¬ (a2 = asing a2)) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow a2) (asing (asing a2))) → ain a1 (aun (asing a1) (apow (asing a2))) → ain a2 (asm a4 (asing a1)) → (¬ ain a0 (asing a3)) → ain a3 (aun a1 (asing a3)) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (asing a2))) → asubq a3 (aun a3 (asing (apow a2))) → asubq a4 (aun (asing (asing a2)) a4) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq a3 (asm (apow a2) (asing (asing a1))) → asubq (aun (asing (asing a1)) (asing (asing (asing a1)))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a2)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun a4 (asing (asm (apow a2) a2))) = a4) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal2 a1) → (¬ aal2 (asing a3)) → (¬ aal4 (asm a4 (asing a1))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun (asing (asing a1)) a4) → aex5 (aun (asing (apow (asing a1))) a4) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25)))
In Proofgold the corresponding term root is 92fcc9... and proposition id is c6d45f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcQaedzSmGabvWFg8Rc9W7FsJyhE6x1123)
(∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → ain a1 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → (¬ ain X27 X25) → ain X27 X26) → asubq (asm X24 X25) X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal2 X25 → aal4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (¬ ain a0 a0) → (¬ ain a1 a1) → (∀X24 : set, asubq X24 (asing a1) → ((X24 = a0) ∨ (X24 = asing a1))) → ain (asing a1) (asing (asing a1)) → (¬ ain a1 (apow (asing a2))) → ain a1 (asm (apow a2) (asing a2)) → (¬ ain a0 (asing (asing a1))) → (¬ asubq (asing a1) (asing a2)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (apow a2)) → (¬ ain a3 (asm a3 (asing a1))) → (¬ (apow (asing a1) = a3)) → (¬ ain (asm a3 (asing a1)) (asm (apow a2) (apow (asing a2)))) → (¬ ain (apow a2) (apow (apow (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain (asing a2) (aun (asing a1) (apow (asing a2))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain a1 (asing a3)) → (¬ ain (asing (asing a1)) a4) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a0 (aun (asing (asing a2)) a4) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a2 (aun a2 (asing (apow a2))) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq a4 (aun (asing (asing a2)) a4) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 a3 → (¬ aal3 (asm a3 (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ aal4 (asm a4 (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)))) → aal2 (apow (asing (asing a1))) → aal5 (aun a4 (asing (apow a2))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (aun a2 (asing (apow a2))) → aex3 (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a1)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))
In Proofgold the corresponding term root is eac35f... and proposition id is 84b4dc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVqjTCJYqsmXjsSYeToDDDR3iHvReFwEfs)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → ain a0 a1 → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → ain a3 a4 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ (X25 = X26)) → aal2 X24) → (∀X24 : set, ∀X25 : set, (¬ aal3 X24) → (¬ aal4 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ (a0 = asing a1)) → ain a1 (asm (apow a2) (apow (asing a2))) → (¬ (a0 = asing (asing a1))) → (¬ (a0 = apow (asing a1))) → (¬ (a2 = asing (asing a1))) → (¬ (a3 = asm (apow a2) a2)) → (¬ ain a0 (asing (apow (asing a1)))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain (asing (asing a1)) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → ain a1 (apow (asm a3 (asing a1))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → (¬ ain a0 (asm a4 a2)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ ain a0 (asing (apow a2))) → (¬ ain a1 (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → ain a0 (aun a4 (asing (apow a2))) → ain a1 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain a1 X24) → asubq X24 (asing (asing a1))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → (¬ asubq (aun (asing (asing a2)) a4) a4) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (asm a2 (asing a1)) a1 → asubq (asm (asm (apow a2) (asing a2)) (asing a1)) (apow (asing a1)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a1)) (asm (apow a2) a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm a4 a2) (asm (asm a4 (asing a1)) (asing a0)) → (asm (asm a4 (apow (asing a2))) (asing a2) = aun (asing a1) (asing a3)) → asubq a2 (aun a3 (asing (asm a3 (asing a1)))) → asubq (asm a3 (asing a1)) a4 → (∀X24 : set, ain X24 a4 → (ain X24 a3 ∨ (X24 = a3))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) a2) → (¬ aal2 (asm (asm (apow a2) a2) (asing X24)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (asm a4 a2) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (asm a4 (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (asing a2)))
In Proofgold the corresponding term root is ce55a7... and proposition id is 351963...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMGW8H6KZSGD9fUdhJi5iibimPm4WRtvWBy)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → ain a0 a3 → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → asubq X25 X26 → asubq X24 X26) → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, (∀X26 : set, ain X26 X24 → (¬ ain X26 X25)) → adisjoint X24 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal5 X24 → aal5 X25) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal4 (aun X24 X25) → (aal2 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (¬ ain a0 a0) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain (asing a1) (asing (asing a1)) → ain a0 (apow (asing a1)) → ain a0 (asm (apow a2) (asing a2)) → ain a1 (apow a2) → (¬ (a0 = asm a3 (asing a1))) → (¬ (a0 = asm (apow a2) a2)) → ain (asing a1) (apow (asing a1)) → (¬ (asing a1 = a2)) → (¬ ain (apow a2) a2) → (¬ (asing a1 = asm a3 (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ asubq (apow a2) (asing a2)) → (¬ (asm a3 (asing a1) = a3)) → (¬ ain (asing a2) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (apow (asing a2))) (apow a2) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asm a3 (asing a1)) (apow (asing a2))) → ain a3 (asing a3) → adisjoint (aun (asing a2) (asing (asing a2))) (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain (asing a1) (asm a4 a2)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain a1 (asing (apow a2))) → (¬ ain (asing a2) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → (X24 = a0)) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = a0)) → ain X24 (aun (asing (asing a1)) (asing (asing (asing a1))))) → asubq (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a2)) → ain X24 (aun a1 (asing a3))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) a4 → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → (¬ aal2 (asm (asm (apow a2) (apow (asing a1))) (asing X24)))) → (¬ aal4 (aun (asing a1) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (apow a2)) → aal4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex2 (aun a1 (asing a3)) → aex2 (aun a1 (asing (apow a2))) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1))))) → aex2 (asm (asm a4 (asing a2)) (asing a3)) → aex3 (asm a4 (asing a1)) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex3 (asm (apow a2) (asing (asing a1))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is 5e2829... and proposition id is 6a50b7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLKrQvm6XiSWeaNck2PMzpbhTwLibfncfQ)
(∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → asubq a1 (asing a0) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex6 X24 → aex5 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (¬ ain a0 (asing a2)) → (¬ (a0 = asm a3 (asing a1))) → (¬ ain a2 (apow (asing a2))) → (¬ ain a0 (asm (apow a2) (apow (asing a2)))) → (¬ ain a1 (asing (asing a1))) → (¬ (asing a1 = asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (¬ ain a3 (apow a2)) → ain (asing (asing a1)) (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (¬ ain a3 (apow (asing a2))) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain a2 (apow (asm a3 (asing a1)))) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a3 (asm a4 (apow (asing a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain a0 (aun (apow (asing a2)) (asing a3)) → ain (apow a2) (asing (apow a2)) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a0 (aun a4 (asing (apow a2))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (asing (asing a1))) → ((X24 = a0) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asm (apow (apow (asing a1))) (asing (apow (asing a1))))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (asm (asm (apow a2) (asing a1)) (asing a0) = asm (apow a2) a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → asubq (asm a4 a2) (asm (asm a4 (apow (asing a2))) (asing a1)) → (asm (asm a4 (apow (asing a2))) (asing a1) = asm a4 a2) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm a4 (asing a3)) a3 → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aal3 (asm a4 (asing a2)) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (asing (asing a1))) a4) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a1)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1))))
In Proofgold the corresponding term root is f70713... and proposition id is e16202...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLrCVnxb3oWZjpEckj119kjcrSo7guyyvn)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, (ain X24 a1 ↔ (X24 = a0))) → (∀X24 : set, (ain X24 a4 ↔ ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = a3)))) → (∀X24 : set, ain X24 a3 → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2))) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal4 X24 → aal4 X25) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex5 X24 → aex4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal7 (aun X24 X25) → (aal6 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (¬ asubq a2 a0) → (¬ ain (asing a1) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq (asing a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (apow a2) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → ain a0 (apow (apow (asing a1))) → ain a1 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (aun (asing (asing a1)) (asing (asing (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asm (apow a2) a2) (asing (asm (apow a2) a2)) → (¬ ain (asing a2) (asing a3)) → ain a0 (aun a1 (asing a3)) → ain a0 (asm a4 (asing a2)) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (apow a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain a0 (aun a1 (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun a4 (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → asubq a2 (aun (asing a1) (apow (asing a2))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (asm (apow a2) a2) (asing (asing a1)) = asing a2) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing a1)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → asubq (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (aun (asing a1) (asing (asing a1)))) → (¬ aal3 (asm (apow a2) a2)) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing a3)) → (¬ aal3 (asm (aun (apow (asing a2)) (asing a3)) (asing X24)))) → (∀X24 : set, ain X24 (apow a2) → (¬ aal4 (asm (apow a2) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asing (asing a2)) a4)) → (¬ aal7 (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aal2 (asm a3 (asing a1)) → aal3 (asm (apow a2) (asing a1)) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aal4 a4 → aex2 (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1)))) → aex5 (aun (asing (apow (asing a1))) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a3)) → (¬ ain (asing a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))))
In Proofgold the corresponding term root is 0c95ec... and proposition id is caa021...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVdzLQLHGs23Wxp1bbaUgwGsQLqYu56G7z)
(∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X25 → (¬ ain X26 X25) → (¬ ain X26 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal6 X24) → (¬ aal5 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, (¬ ain X27 X26) → asubq X26 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (¬ asubq a4 a3) → (∀X24 : set, asubq X24 a2 → (¬ ain a1 X24) → asubq X24 a1) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a0 (apow (asing a1)) → (¬ asubq a1 (asing a1)) → (¬ ain a1 (asing a2)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ asubq (asing a1) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) a2) → (¬ asubq (asm (apow a2) (apow (asing a2))) a2) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asing a2)) → (¬ ain (asing (asing a1)) a3) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → (¬ ain a1 (asing (apow (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a3 (aun (asing a1) (apow (asing a2)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain a0 (asing a3)) → (¬ ain (asing (asing a1)) a4) → ain a3 (asm a4 a2) → ain a3 (asm a4 (asing a1)) → ain (asing a2) (aun (apow (asing a2)) (asing a3)) → ain a0 (aun a2 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → ain a1 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq (asm a2 (asing a0)) (asing a1) → (asm a2 (asing a0) = asing a1) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq (asm a4 (apow (asing a2))) (asm a4 (asing a0)) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal4 (asm (apow a2) (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → aex3 (aun (asing a2) (apow (asing a2))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex5 (aun a4 (asing (asm (apow a2) a2))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24))))
In Proofgold the corresponding term root is 76ffce... and proposition id is d320f0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMP2UUB5bvRhUyb66TSapVbYKcszKhnkww6)
(∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → ain a0 a1 → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → (¬ ain X26 X25) → ain X26 (asm X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (a1 = asing a0) → (∀X24 : set, aal4 X24 → aal3 X24) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal5 X24 → aal4 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a2 a0) → (¬ ain a2 a2) → asubq a3 a4 → (¬ (a0 = a3)) → (¬ (a2 = a3)) → asubq a1 (asm a3 (asing a1)) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ asubq (apow a2) (asm (apow a2) (apow (asing a2)))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing a1) (apow (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain a1 (aun a3 (asing (asing a2))) → (¬ ain (apow a2) (aun a3 (asing (asing a2)))) → (¬ ain a0 (asing a3)) → (¬ ain (asm a3 (asing a1)) a4) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a1 (asm a4 (apow (asing a2))) → ain (asing a2) (aun (asing (asing a2)) a4) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm a3 (asing a1)) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain (apow a2) (aun a2 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a4 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (apow (asing a1)) (asm (asm (apow a2) (asing a2)) (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing a1)) → ain X24 (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (aun (asing a2) (apow (asing a2))) (aun (apow a2) (asing (asing a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) = aun a3 (asing (asing a2))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (asm a3 (asing a1)) (aint (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing a1))) → aal2 (asm (apow a2) a2) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex4 (asm (aun a4 (asing (apow a2))) (asing a0)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (¬ (a1 = apow (asing a1)))
In Proofgold the corresponding term root is 48b332... and proposition id is 36999a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNn3tXG3qKAkkjdJpDkYZR28DimhS9g72K)
(∀X24 : set, (aal6 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal5 X25))))) → (∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (ain X24 a3 ↔ (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (¬ aal6 X24) → (¬ aal7 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a0 a0) → (¬ ain a2 a2) → (¬ asubq a2 a0) → (¬ (a0 = asing a1)) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asm a3 (asing a1)) a2) → (¬ (a2 = apow (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → asubq (aun (asing a1) (asing (asing a1))) (asm (apow a2) (asing a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → (¬ (a2 = asing a2)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain a0 (apow (asing (asing a1))) → ain a0 (apow (apow (asing a1))) → (¬ ain (asing (asing a1)) (asing (apow (asing a1)))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a0 (apow (aun (asing a1) (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ain a2 (asm a4 (asing a1)) → (¬ ain a1 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a1 (aun (asing (asing a1)) (asing a3))) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a2 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → (¬ ain (asm a3 (asing a1)) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a2 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, ain X24 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (apow a2) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (¬ asubq (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing (asing a2)) a4)) → asubq a1 (asm a2 (asing a1)) → asubq a2 (asm a3 (asing a2)) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → (asm (asm (apow a2) (apow (asing a2))) (asing a2) = aun (asing a1) (asing (asing a1))) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a1)) (asing a3) = asm a3 (asing a1)) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (¬ aal2 (asing a3)) → (∀X24 : set, ain X24 a2 → (¬ aal2 (asm a2 (asing X24)))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal3 (asm (aun (apow (asing a2)) (asing (apow a2))) (asing X24)))) → aal2 (asm a4 a2) → aal3 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → aal3 (asm a4 (asing a2)) → aal3 (aun a2 (asing (apow a2))) → aal4 (apow a2) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal5 (aun (apow a2) (asing (asing a2))) → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (asm a3 (asing a1)) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (aun a2 (asing (apow a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing X24)))) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1))))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun a4 (asing (apow a2))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))))
In Proofgold the corresponding term root is 9dba07... and proposition id is f922ea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMYybWCv1uPCo2fEkjPUQGXbYK7tFrTE6Se)
(∀X24 : set, ∀X25 : set, (asubq X24 X25 ↔ (∀X26 : set, ain X26 X24 → ain X26 X25))) → (∀X24 : set, (aal7 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal6 X25))))) → (∀X24 : set, (aex5 X24 ↔ (aal5 X24 ∧ (¬ aal6 X24)))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → ain a2 a3 → (∀X24 : set, ∀X25 : set, asubq X25 X24 → ain X25 (apow X24)) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq X26 X24 → asubq X26 X25 → (aint X24 X25 = X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → (∀X24 : set, ∀X25 : set, (¬ ain X25 X24) → aal2 X24 → aal3 (aun X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 X24 → aal3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal6 (aun X24 X25) → (aal5 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → (aal6 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → ain a1 X24 → (X24 = a2)) → ain a1 (asm (apow a2) (asing a2)) → (¬ (a1 = asing a2)) → (¬ ain a1 (asm (apow a2) a2)) → (¬ (a2 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → (¬ ain a0 X24) → (X24 = a0)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a2))) → ain (asing (asing a1)) (apow (apow (asing a1))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → ain a0 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a1 (aun a3 (asing (asing a2))) → ain (asing a2) (aun a3 (asing (asing a2))) → (¬ ain a2 (apow (asm a3 (asing a1)))) → ain a3 (asing a3) → (¬ ain (asing (asing a1)) a4) → (¬ ain (asing a2) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun a1 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a2 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → asubq a2 (aun (asing a1) (apow (asing a2))) → (¬ asubq (asm a4 (asing a2)) a2) → (¬ asubq (aun a3 (asing (asm (apow a2) a2))) a3) → asubq a4 (aun (asing (asing a1)) a4) → (¬ asubq (aun (asing (asing a1)) a4) a4) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq a2 (asm a3 (asing a2)) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq a1 (asm (asm a3 (asing a1)) (asing a2)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) (apow (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) (apow a2) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → asubq a2 (asm (asm a4 (asing a2)) (asing a3)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → asubq (asm a4 (asing a3)) a3 → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2))) = asm (apow a2) (apow (asing a1))) → (¬ aal3 (aun (asing (asing a1)) (asing (asing (asing a1))))) → (¬ aal5 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal4 (asm (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) (asing X24)))) → aal2 (asm (apow a2) (apow (asing a1))) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (asm (asm a4 (asing a1)) (asing a0)) → (¬ aal3 (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex3 (asm (aun a3 (asing (apow a2))) (asing a1)) → aex4 (asm (aun (asing (asing a2)) a4) (asing a2)) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3)) (aun (asing a1) (apow (asing a2))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → aex5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex3 (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)))
In Proofgold the corresponding term root is 45a3bf... and proposition id is d246b3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMSAv3dfqRFxMNmo62bfNSkvbh69XP5aafk)
ain a2 a3 → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → adisjoint X25 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal7 X24) → (¬ aal6 (asm X24 (asing X25)))) → asubq (asing a0) a1 → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ ain a0 a0) → (¬ ain a2 a1) → asubq a3 a4 → (¬ (a1 = a2)) → (∀X24 : set, asubq X24 a2 → ain a0 X24 → (¬ ain a1 X24) → (X24 = a1)) → ain a2 (asing a2) → (¬ asubq a1 (asing a1)) → ain a0 (asm (apow a2) (asing a1)) → (¬ ain (asing a2) a1) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a2 (apow (asing a1))) → asubq (asing a1) (asm (apow a2) (asing a2)) → (¬ (a2 = asing (asing a1))) → (¬ (asing (asing a1) = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain a0 X24) → asubq X24 (asing (asing a1))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ ain (apow (asing a1)) a3) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a0 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a2 (aun a1 (asing a3))) → (¬ ain (apow (asing a1)) a4) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain a2 (asm a4 a2) → ain a1 (asm a4 (apow (asing a2))) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain a0 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → (¬ ain a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun a4 (asing (apow a2))) → (¬ ain (asing a1) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → asubq a3 (aun a3 (asing (asm (apow a2) a2))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (¬ asubq (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a2))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → (asm a3 (asing a0) = asm (apow a2) (apow (asing a1))) → asubq (asing a2) (asm (asm a3 (asing a1)) (asing a0)) → (asm (asm a3 (asing a1)) (asing a0) = asing a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a1)) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1)) = asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → asubq (asm (asm a4 a2) (asing a2)) (asing a3) → (asm (asm a4 (asing a1)) (asing a2) = aun a1 (asing a3)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq a2 (apow (asm a3 (asing a1))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing (asm a3 (asing a1)))) a2 → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal3 a2) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → (¬ aal2 (asm (asm a3 (asing a1)) (asing X24)))) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal4 (aun (apow (asing a2)) (asing (apow a2)))) → (¬ aal5 a4) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a0))) → aal3 (asm (apow a2) (apow (asing a2))) → aal4 (aun a3 (asing (apow (asing a1)))) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → (¬ aal4 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1))))) → aex4 (aun a3 (asing (asing a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a3))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a2)) → ain X24 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (asing a2)))) → (∀X24 : set, ain a0 (apow X24))
In Proofgold the corresponding term root is c37b71... and proposition id is 27019b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcmJnXFK7CHCLbTZg2gyjmpePdyqcnapqU)
(∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aun X24 X25) ↔ (ain X26 X24 ∨ ain X26 X25))) → ain a0 a4 → (∀X24 : set, asubq X24 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq (aun X24 X25) (aun X24 X26) → asubq X25 X26) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal5 X24) → (¬ aal4 (asm X24 (asing X25)))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → (∀X26 : set, ain X26 X25 → aal3 (aun X24 (asing X26)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → aal4 (aun (asm X24 (asing X27)) X25))) → ain a0 (asm a3 (asing a1)) → (¬ ain a0 (aun (asing a1) (asing (asing a1)))) → (¬ ain a0 (asm (apow a2) a2)) → ain a2 (asm (apow a2) (asing a1)) → (¬ (a1 = asing a2)) → (¬ (a1 = apow (asing a1))) → (¬ asubq (asing a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (¬ ain (apow (asing a1)) a3) → (¬ ain a3 (asm a3 (asing a1))) → asubq (asm a3 (asing a1)) (apow a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a1))) → ((X24 = a1) ∨ (X24 = a2))) → (¬ (asm a3 (asing a1) = a3)) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → (¬ ain (apow a2) (apow (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (apow a2) (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → ain a3 (asm a4 a2) → (¬ ain (apow a2) (aun (asing (asing a2)) a4)) → ain (apow a2) (asing (apow a2)) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → (¬ ain (asing a1) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq a2 (aun a2 (asing (apow a2))) → (¬ asubq (aun a3 (asing (apow (asing a1)))) a3) → asubq a3 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (apow a2))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → asubq (asm a3 (asing a0)) (asm (apow a2) (apow (asing a1))) → asubq (asing a2) (asm (asm (apow a2) (apow (asing a1))) (asing a1)) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0) = aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a0)) → ain X24 (asm a4 (apow (asing a2)))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → (¬ aal3 (apow (asing a2))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → aal2 a2 → aal2 (asm a3 (asing a1)) → aal5 (aun (asing (apow (asing a1))) a4) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (aun (asing a2) (apow (asing a2))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aal4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a1))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a1))) → (¬ aal4 a3)
In Proofgold the corresponding term root is a92c37... and proposition id is 061854...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMXUdrjKZ3B2JLzih2HYxJq4QEiHYxnVDPB)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ain X24 X25 → aal2 (apow X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal3 X26 → aal3 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal4 X24 ∨ aal2 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal6 X26 → aal6 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 a2 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → ain a2 (asing a2) → (¬ asubq a2 (asing a1)) → (¬ asubq (asm a3 (asing a1)) a2) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → asubq (asm (apow a2) a2) (asm (apow a2) (apow (asing a2))) → (¬ (asm a3 (asing a1) = apow a2)) → ain (asing (asing a1)) (asing (asing (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (apow (asing a1)) (aun a3 (asing (apow (asing a1)))) → (¬ ain (asm (apow a2) a2) (asing (asing a2))) → ain a1 (asm a4 (asing a2)) → adisjoint a2 (aun (asing (asing a1)) (asing a3)) → ain (asing a1) (aun (asing (asing a1)) a4) → ain a3 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a0 (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm (apow a2) a2) (aun a4 (asing (apow a2)))) → ain a3 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → (X24 = asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → (¬ asubq (aun a2 (asing (apow a2))) a2) → asubq a4 (aun (asing (asing (asing a1))) a4) → (¬ asubq (aun (asing (apow (asing a1))) a4) a4) → asubq (aun (asing (asing a2)) a4) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (asm (asm (apow a2) (asing a1)) (asing (asing a1))) (asm a3 (asing a1)) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2)) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing a3))) → asubq (asm (asm a4 (apow (asing a2))) (asing a3)) (asm (apow a2) (apow (asing a1))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → asubq (aun (asing a2) (apow (asing a2))) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2))) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal6 (aun (asing (asing a2)) a4)) → aal5 (aun a4 (asing (asm (apow a2) a2))) → aex2 (asm a3 (asing a1)) → aex2 (asm (apow a2) a2) → aex2 (aun (asing a2) (asing (asing a2))) → aex3 (asm a4 (asing a1)) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (aun a3 (asing (apow (asing a1)))) → aex3 (asm a4 (asing a0)) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing a0)) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → ain (asing a1) (apow a2)
In Proofgold the corresponding term root is 4b12df... and proposition id is d66524...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMK1DH6w1JeN2wUWngA2rVMscgbCkyBcDZp)
ain a2 a3 → (∀X24 : set, asubq a0 X24) → (∀X24 : set, ain X24 (apow X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (aun X24 (aun X25 X26) = aun (aun X24 X25) X26)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → asubq (asing X25) X24) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (¬ ain a2 a2) → asubq a2 a3 → (¬ asubq a2 a0) → ain a1 (aun (asing a1) (asing (asing a1))) → (¬ asubq a1 (asing a1)) → (¬ ain a1 (apow (asing a1))) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain (apow a2) a2) → (¬ asubq (asing a2) a2) → (¬ ain a2 (asing (asing a1))) → (∀X24 : set, ain (asing a1) X24 → ain a0 X24 → asubq (apow (asing a1)) X24) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ asubq (apow a2) (apow (asing a1))) → (¬ ain (asing (asing a1)) a3) → (¬ (asing (asing a1) = a3)) → (∀X24 : set, ain X24 (asm (apow a2) a2) → ((X24 = asing a1) ∨ (X24 = a2))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (¬ ain (asing a1) (asing (asing a2))) → ain a2 (aun (asing a2) (apow (asing a2))) → (¬ ain (asing (asing a1)) a4) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain a1 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain (apow a2) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (apow (aun (asing a1) (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → asubq a3 (aun a3 (asing (asing a2))) → (¬ asubq (aun (asing (asing a1)) a4) a4) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq a4 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (asm a4 (asing a1)) → (¬ asubq (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (apow (asing (asing a1)))) → (¬ asubq (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, ain X24 a3 → (¬ (X24 = a2)) → ain X24 a2) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → asubq (apow (asing a1)) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1)))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a3)) → ain X24 (asm a3 (asing a1))) → (asm a4 (asing a3) = a3) → asubq (aun (asing a2) (apow (asing a2))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aun (asing (asing a2)) a4) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (∀X24 : set, ain X24 (aun (apow a2) (asing (asing a2))) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2))))) → (¬ aal6 (aun (asing (asing a1)) a4)) → aal5 (aun (asing (apow (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → aex2 a2 → aex2 (aun (asing a1) (asing (asing a1))) → aex2 (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) → aex3 (asm (apow a2) (asing a2)) → aex2 (asm a3 (asing a2)) → aex4 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → aex4 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a0)) → aex5 (aun (asing (asing a1)) a4) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → (¬ asubq (asing a1) (asing a2))
In Proofgold the corresponding term root is dcc79e... and proposition id is caee36...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMTaczw4hdtCHMWzuULTdXzF3KJWEGwtx1x)
(∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (asm X24 X25) ↔ (ain X26 X24 ∧ (¬ ain X26 X25)))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → (ain X26 X24 ∨ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X24 X26 → asubq X25 X26 → asubq (aun X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X25 → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal4 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal4 X24 → aal2 X25 → aal6 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X25 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (¬ asubq a2 a1) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → ain a0 (apow a2) → (¬ ain a1 (apow (asing a1))) → (¬ ain (asing a1) (asing a2)) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) (asm (apow a2) (apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ain a0 X24 → (X24 = apow (asing a1))) → (¬ (apow (asing a1) = a3)) → (¬ ain a3 (asm (apow a2) (asing a1))) → (¬ (asm (apow a2) a2 = apow a2)) → ain (asing a1) (aun (asing (asing a1)) (asing (asing (asing a1)))) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ ain (asing a1) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → ain a0 (asm a4 (asing a1)) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain (asing a2) (aun (asing a2) (apow (asing a2))) → ain a0 (aun a1 (asing a3)) → (¬ ain (asm (apow a2) a2) a4) → (¬ ain a0 (aun (asing (asing a1)) (asing a3))) → ain a2 (asm a4 (apow (asing a2))) → ain a3 (aun (apow (asing a2)) (asing a3)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain (asing a2) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → ain a3 (aun (asing (asing a2)) a4) → ain a2 (aun (asing (asing a2)) a4) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) a3) → asubq a3 (aun a3 (asing (apow (asing a1)))) → asubq a3 (aun a3 (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (asm (apow a2) a2) (asing (asing a1))) (asing a2) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing a2)) (aun (asing a1) (asing (asing a1))) → (asm (apow a2) (asing (asing a1)) = a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a2) = aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → (asm (asm a4 (apow (asing a2))) (asing a3) = asm (apow a2) (apow (asing a1))) → (asm a4 (asing a0) = asm a4 (apow (asing a2))) → asubq (asm a3 (asing a1)) (aun a4 (asing (apow a2))) → asubq (asm (apow a2) (apow (asing a1))) a4 → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → asubq (asm a3 (asing a1)) (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (¬ aal3 (aun a1 (asing a3))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ aal3 (asm (asm (apow a2) (asing a2)) (asing X24)))) → (¬ aal4 (asm (apow a2) (apow (asing a2)))) → (¬ aal5 (apow a2)) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = a1))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aex3 (aun a2 (asing (apow a2))) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex4 (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (apow (asing a1))) a4) → aex5 (aun a4 (asing (asm (apow a2) a2))) → aex4 (asm (aun a4 (asing (apow a2))) (asing a3)) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a1)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (aun (apow (asing a2)) (asing a3))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ ain X24 (asing (asing a2))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (asing a2))) (aun a3 (asing (apow a2))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1))))
In Proofgold the corresponding term root is 470e77... and proposition id is c28bdf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMLmxKhbBbYvVk3TAb2uYzKQ2z4VJ1PUDkP)
(∀X24 : set, (aex3 X24 ↔ (aal3 X24 ∧ (¬ aal4 X24)))) → (∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (asing X24) ↔ (X25 = X24))) → (∀X24 : set, ain X24 a1 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 (asm X24 X25))) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X24 → aal3 X25 → aal5 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 X24 → aal2 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex4 X24 → aex3 (asm X24 (asing X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal3 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aex5 X24 → aex2 (aint X24 X25) → aex3 (asm X24 X25)) → (¬ ain a2 a2) → asubq a2 a3 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → asubq X24 (asing a1)) → (¬ ain a1 (apow (asing a2))) → (¬ (a0 = asing a2)) → (¬ (a0 = asm (apow a2) a2)) → ain a2 (apow a2) → ain (asing a1) (asm (apow a2) (asing a1)) → ain (asing a1) (asm (apow a2) (apow (asing a2))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ ain (asing (asing a1)) a2) → (¬ ain (asing a2) a2) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ain a0 X24 → (X24 = a1)) → (∀X24 : set, ain X24 (asm a3 (asing a1)) → ((X24 = a0) ∨ (X24 = a2))) → (¬ (asing a2 = asm (apow a2) a2)) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (a3 = apow a2)) → (¬ ain (apow a2) (apow (apow (asing a1)))) → (¬ ain (asing a1) (asing (aun (asing a1) (asing (asing a1))))) → ain a1 (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → (¬ ain (apow a2) (aun (apow a2) (asing (asing a2)))) → ain (asm a3 (asing a1)) (asing (asm a3 (asing a1))) → (¬ ain (asing a2) (aun a1 (asing a3))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (asing a2) (aun (asing (asing a2)) (asing (apow a2))) → ain a2 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (asing (asing (asing a1)))) → ((X24 = asing a1) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → (¬ asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) a2) → asubq a4 (aun a4 (asing (apow a2))) → (¬ asubq (asm a4 (asing a1)) (asm a3 (asing a1))) → (¬ asubq (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1))))) → asubq (asm (apow a2) (apow (asing a1))) (asm a3 (asing a0)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (asing a2)) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a1)) → ain X24 (apow (asing a1))) → asubq (asm (asm (apow a2) (asing a2)) (asing (asing a1))) a2 → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = asing a1)) → ain X24 (asm a3 (asing a1))) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a2)) (asing a1)) → (asm (asm a4 (asing a1)) (asing a0) = asm a4 a2) → asubq (asm a3 (asing a1)) (aun (apow a2) (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1)) = asm a3 (asing a1)) → asubq (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (¬ aal3 (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (apow (asing a2))) → (¬ aal3 (asm (asm a4 (apow (asing a2))) (asing X24)))) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (∀X24 : set, aal5 (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing X24)) → (¬ (X24 = asing (asing a1)))) → (¬ aal6 (aun (asing (asing (asing a1))) a4)) → aal2 (aun (asing a1) (asing (asing a1))) → aal3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aal4 (apow a2) → aex2 (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex2 (aun (asing a3) (asing (apow a2))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a0)) (aun (asing a1) (asing (asm a3 (asing a1)))) → aex4 (apow a2) → aex4 (aun (apow (asing a1)) (asm a4 a2)) → aex4 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → aex3 (asm (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) (asing a0)) → aex6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (asing a2)) → (¬ ain a1 a1)
In Proofgold the corresponding term root is b86aa1... and proposition id is 43ad37...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMEvfuiirtP6gCmM2A6iRG1c6onooSCxoEr)
(∀X24 : set, (aex4 X24 ↔ (aal4 X24 ∧ (¬ aal5 X24)))) → ain a1 a3 → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (asm X24 X25) → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, asubq (aint X24 X25) X25) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 X25 → asubq (asm X24 X25) (asm X24 X26)) → (∀X24 : set, ∀X25 : set, (¬ ain X25 (asing X24)) → (¬ (X25 = X24))) → (a1 = asing a0) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → aal5 (aun (asm X24 (asing X27)) X25))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal5 X26 → (aal5 X24 ∨ aal2 X25))) → asubq a2 a3 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → ain a1 X24 → (X24 = asing a1)) → (¬ (a1 = asing a2)) → (¬ ain (asing a2) a2) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = a3)) → (¬ (a2 = asing (asing a1))) → (¬ ain (asing a1) a3) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → asubq X24 a1) → (¬ asubq (asm a3 (asing a1)) (asing a2)) → (¬ asubq (apow a2) (asing a2)) → (¬ (asing a2 = asm a3 (asing a1))) → ain (apow (asing a1)) (asing (apow (asing a1))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) a4) → (¬ ain (asm (apow a2) a2) (aun (apow a2) (asing (asing a2)))) → (¬ ain (asing a1) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → ((X24 = a2) ∨ (X24 = a3))) → (∀X24 : set, ain X24 (asm a4 a2) → ((X24 = a2) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow a2))) → (¬ asubq (aun (asing (asing a2)) a4) a4) → asubq (asm a2 (asing a0)) (asing a1) → asubq (asm a2 (asing a1)) a1 → (asm a3 (asing a2) = a2) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (asm (asm (apow a2) a2) (asing a2) = asing (asing a1)) → (asm (asm (apow a2) (asing a1)) (asing a2) = apow (asing a1)) → asubq (aun (asing a1) (asing (asing a1))) (asm (asm (apow a2) (apow (asing a2))) (asing a2)) → asubq (apow (asing (asing a1))) (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing a1))) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing (asing (asing a1))) = apow (asing a1)) → asubq (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing a1))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a3)) → ain X24 (asing a2)) → asubq (asm (asm a4 (asing a1)) (asing a2)) (aun a1 (asing a3)) → asubq (asm (asm a4 (apow (asing a2))) (asing a1)) (asm a4 a2) → asubq (aun (asing a2) (apow (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (aint (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)) = asm a3 (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ aal3 (asm (asm (apow a2) (apow (asing a2))) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → (¬ aal3 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 (asm (apow a2) a2) → aal2 (asm a4 a2) → aal3 (asm a4 (asing a1)) → aal4 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → aex2 (apow (asing a1)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → (¬ ain X24 (asing a0)) → ain X24 (aun (asing a1) (asing (asm a3 (asing a1))))) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex5 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → aex5 (aun a4 (asing (apow a2))) → aex3 (asm (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) → aex3 (asm (aun a4 (asing (apow a2))) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a2)) → aex5 (asm (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (asing a0)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (¬ aal6 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))))) → asubq (asm (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) (asing a1)) (aun (asing a0) (asing (asm a3 (asing a1))))
In Proofgold the corresponding term root is 09e40d... and proposition id is 3977ab...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMcA13wK8EtQboRn8q9f9yk3H17GgQJkNHZ)
(∀X24 : set, ∀X25 : set, ain X24 X25 → (¬ ain X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aint X24 X25) → ain X26 X24) → (∀X24 : set, ∀X25 : set, (X24 = aun (asm X24 X25) (aint X24 X25))) → (∀X24 : set, ∀X25 : set, aal3 (aun X24 X25) → (aal2 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal4 (asm X24 (asing X25)) → aal5 X24) → asubq a2 a3 → (∀X24 : set, asubq X24 a2 → (¬ ain a0 X24) → (¬ ain a1 X24) → (X24 = a0)) → (¬ (a0 = asing a1)) → (¬ (a0 = asing a2)) → ain (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a0 (asm (apow a2) a2)) → (¬ ain a3 (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → (¬ ain a1 (asm a3 (asing a1))) → (∀X24 : set, ain X24 (asing (asing a1)) → (X24 = asing a1)) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → (X24 = asing (asing a1))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → (¬ ain a0 X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → ain (asing a1) X24 → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (∀X24 : set, asubq X24 (apow (asing a1)) → (¬ ain (asing a1) X24) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = apow (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ (a1 = apow a2)) → (¬ (a3 = apow a2)) → (¬ (asm (apow a2) (apow (asing a2)) = apow a2)) → ain (asing (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → ain (asing (asing a1)) (aun (asing (asing (asing a1))) a4) → ain (apow (asing a1)) (aun (asing (apow (asing a1))) a4) → ain a3 (aun (apow (asing a2)) (asing a3)) → (¬ ain a2 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3)))) → (¬ ain (asm a3 (asing a1)) (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → ain (apow a2) (aun (apow a2) (asing (apow a2))) → ain (apow a2) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → ain (asing a2) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → ain (asing a1) X24 → ain a1 X24 → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a2)) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a3))) → asubq a3 (aun a3 (asing (apow (asing a1)))) → (¬ asubq (aun a4 (asing (asm (apow a2) a2))) a4) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (asm (apow a2) (apow (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → asubq (asm (asm (apow a2) a2) (asing a2)) (asing (asing a1)) → asubq (asm (asm a4 (asing a2)) (asing a3)) a2 → (∀X24 : set, ain X24 (apow a2) → (ain X24 a3 ∨ (X24 = asing a1))) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a3)) → (¬ aal3 (aun (asing a1) (asing a3))) → (¬ aal3 (asm a4 a2)) → (¬ aal5 (aun a3 (asing (asm (apow a2) a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal6 (aun (asing (apow (asing a1))) a4)) → aal2 (apow (asing (asing a1))) → aal3 a3 → aal4 (aun a3 (asing (apow (asing a1)))) → aal5 (aun (asing (asing a1)) a4) → aex3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex4 (aun a2 (aun (asing (asing a1)) (asing a3))) → aex4 (aun a2 (aun (asing a3) (asing (apow a2)))) → aex4 (aun a3 (asing (asm (apow a2) a2))) → aex4 (aint (aun a4 (asing (asm (apow a2) a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))) → aex5 (aun (asing (apow (asing a1))) a4) → aex6 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a2)) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ (a2 = asing a2))
In Proofgold the corresponding term root is 217b9a... and proposition id is e4b22f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMFDr1VXHBfxpZz2Fabw9RAtFoYtXJbukWY)
(∀X24 : set, (aal4 X24 ↔ (∃X25 : set, (asubq X25 X24 ∧ ((¬ asubq X24 X25) ∧ aal3 X25))))) → (∀X24 : set, (aex6 X24 ↔ (aal6 X24 ∧ (¬ aal7 X24)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq (aun (aun X24 X25) X26) (aun X24 (aun X25 X26))) → (∀X24 : set, ∀X25 : set, asubq X25 X24 → (¬ asubq X24 X25) → (∃X26 : set, (ain X26 X24 ∧ asubq X25 (asm X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, asubq X24 (apow (asing X25)) → aal2 X24 → asubq (apow (asing X25)) X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (aint X24 (asing X25) = asing X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aex3 X24 → aex2 (asm X24 (asing X25))) → (¬ (a1 = a3)) → ain a0 (apow a2) → (¬ (a0 = asing a1)) → (¬ (a0 = asm a3 (asing a1))) → (¬ (a1 = asm a3 (asing a1))) → ain (asing a1) (asm (apow a2) (asing a1)) → (¬ ain (asing a1) (apow (asing a2))) → asubq (asing a1) (aun (asing a1) (asing (asing a1))) → (¬ ain a1 (asm a3 (asing a1))) → (¬ (a0 = aun (asing a1) (asing (asing a1)))) → (¬ asubq (asing a2) a2) → (¬ ain (asm (apow a2) a2) (asing (asing a1))) → (¬ ain (asing a1) (asm a3 (asing a1))) → (¬ ain a2 (aun (asing a1) (asing (asing a1)))) → (¬ asubq (asm (apow a2) (asing a2)) (aun (asing a1) (asing (asing a1)))) → (¬ (a2 = asm a3 (asing a1))) → (¬ asubq (apow a2) (asing a2)) → (¬ ain (asing a2) (asm (apow a2) a2)) → (¬ ain a3 (asm (apow a2) (asing a1))) → ain a0 (apow (apow (asing a1))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a0 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ain (asing a1) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → (¬ ain (apow a2) (apow (asing a2))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → (¬ ain a0 (asing a3)) → adisjoint (apow (asing a1)) (asm a4 a2) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain (apow a2) (aun a1 (asing (apow a2))) → ain a1 (aun a2 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → ain (apow a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asing a1) (asing (asing a1))) → ((X24 = a1) ∨ (X24 = asing a1))) → (∀X24 : set, ain X24 (asing (asing (asing a1))) → (X24 = asing (asing a1))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → ((((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (apow a2) (aun (apow a2) (asing (asing a2))) → asubq (apow (asing (asing a1))) (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → asubq (asing a1) (asm a2 (asing a0)) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a1) = apow (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = asing a1)) → ain X24 a2) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = a0)) → ain X24 (asm (apow a2) (apow (asing a2)))) → asubq a3 (asm (apow a2) (asing (asing a1))) → (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1))) = apow a2) → asubq (asm (asm a4 (asing a2)) (asing a0)) (aun (asing a1) (asing a3)) → asubq (aun (asing a1) (asing a3)) (asm (asm a4 (asing a2)) (asing a0)) → (asm (asm a4 (asing a2)) (asing a0) = aun (asing a1) (asing a3)) → (asm (asm a4 (asing a2)) (asing a3) = a2) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm a3 (asing a1)) (asm (asm a4 (asing a1)) (asing a3)) → asubq a3 (aun a4 (asing (asm (apow a2) a2))) → (∀X24 : set, ain X24 a3 → ain X24 (apow (asm a3 (asing a1))) → ain X24 a2) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain X24 (asing (apow a2)))) → (¬ aal4 (asm (apow a2) (asing a1))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (¬ aal3 (asm (asm a4 (asing a1)) (asing X24)))) → (¬ aal5 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → (¬ aal5 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal5 (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2))))) → aal2 a2 → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun a3 (asing (apow a2))) → aal3 (aint (aun a3 (asing (asm a3 (asing a1)))) (apow (asm a3 (asing a1)))) → aex2 (asm a3 (asing a1)) → aex2 (asm (apow a2) (apow (asing a1))) → aex2 (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → aex2 (asm (asm a4 (asing a1)) (asing a2)) → aex5 (aun a4 (asing (apow a2))) → aex6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 a4 → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing X24)))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing a0)) (aun (asm (apow a2) (apow (asing a1))) (aun (asing (asing a2)) (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → ain X24 (aun a3 (asing (apow a2)))) → ain a2 (aun (asing a2) (apow (asing a2)))
In Proofgold the corresponding term root is ff4bfe... and proposition id is 7b2e2e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMNbaJdBt5CyS34rjfZVJxFckmNQpGpkBQZ)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, (¬ ain X24 a0)) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ain X24 (asing X24)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ asubq X24 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (¬ ain X26 X24) → (¬ ain X26 X25) → (¬ ain X26 (aun X24 X25))) → asubq a1 (asing a0) → (∀X24 : set, (¬ aal2 (asing X24))) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal4 X24 ∨ aal2 X25)) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ain X25 X24 → aal3 (asm X24 (asing X25)) → aal4 X24) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aex2 X24 → aex2 X25 → aex4 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal3 X24 ∨ aal3 X25)) → (¬ ain a0 (asing a1)) → (¬ ain a1 (apow (asing a1))) → (¬ asubq (asing a1) (asing a2)) → (¬ asubq (asm (apow a2) a2) a2) → (¬ ain (asing a2) (asing (asing a1))) → (¬ (asing a1 = a3)) → asubq (asing (asing a1)) (apow (asing a1)) → asubq (asing (asing a1)) (aun (asing a1) (asing (asing a1))) → (¬ ain (apow a2) (asing a2)) → (¬ asubq a3 (asing a2)) → asubq (asm a3 (asing a1)) (apow a2) → (¬ asubq (asm (apow a2) (asing a1)) (asm (apow a2) a2)) → (¬ (asm a3 (asing a1) = apow a2)) → (¬ ain (asing a1) (apow (asing (asing a1)))) → ain (asing (asing a1)) (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a2))) (asing (asing (asing a1)))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → (¬ ain a3 (aun (asing a2) (apow (asing a2)))) → ain a0 (aun a3 (asing (asing a2))) → (¬ ain (asing a1) (asm a4 a2)) → (¬ ain (asm (apow a2) a2) (asing (apow a2))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ ain a0 (aun (asing a3) (asing (apow a2)))) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → asubq X24 (asing a1)) → (∀X24 : set, asubq X24 (aun (asing a1) (asing (asing a1))) → (¬ ain (asing a1) X24) → (¬ ain a1 X24) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = asing (asing a1))) ∨ (X24 = aun (asing a1) (asing (asing a1))))) → (∀X24 : set, ain X24 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → (((X24 = a0) ∨ (X24 = a1)) ∨ (X24 = asing (asing a1)))) → (∀X24 : set, ain X24 a4 → (¬ (X24 = a1)) → (((X24 = a0) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a4 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → (¬ asubq (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) (apow (asing (asing a1)))) → asubq (asm (apow a2) (asing a1)) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (asm (asm a3 (asing a1)) (asing a2)) a1 → asubq (asing a2) (asm (asm (apow a2) a2) (asing (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ (X24 = a2)) → ain X24 (apow (asing a1))) → (asm (asm (apow a2) (apow (asing a2))) (asing a1) = asm (apow a2) a2) → (∀X24 : set, ain X24 (apow a2) → (¬ (X24 = asing a1)) → ain X24 a3) → (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0) = aun (asing (asing a1)) (asing (asing (asing a1)))) → (∀X24 : set, ain X24 (asm a4 a2) → (¬ (X24 = a2)) → ain X24 (asing a3)) → asubq (aun a1 (asing a3)) (asm (asm a4 (asing a1)) (asing a2)) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm a4 a2)) → asubq (asm (apow a2) (apow (asing a1))) (aun a4 (asing (apow a2))) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) = apow a2) → asubq (aint (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) (asm a3 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal4 (aun (asing a2) (apow (asing a2)))) → (¬ aal5 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2)))))) → (¬ aal5 (aun a3 (asing (apow a2)))) → (¬ aal5 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → aal3 (aun (asing a2) (apow (asing a2))) → aal4 (aun a3 (asing (asm (apow a2) a2))) → aal4 (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → aal6 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → aal5 (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) → aex2 (aint (aun a4 (asing (apow a2))) (asm (apow a2) (apow (asing a2)))) → aex2 (asm (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) (asing a0)) → aex3 (asm a4 (asing a1)) → aex3 (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → aex3 (asm (aun (asing (asing a1)) a4) (aun (asing a2) (apow (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ain X24 (apow (asing a2)) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → aal6 X24 → aal6 X25)
In Proofgold the corresponding term root is b2ba6f... and proposition id is 886033...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMbuKvCwqRoTZ9mr2usyFBFJnjtnbvgeoJV)
(∀X24 : set, (aex2 X24 ↔ (aal2 X24 ∧ (¬ aal3 X24)))) → (∀X24 : set, ∀X25 : set, asubq X24 X25 → asubq X25 X24 → (X24 = X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 X24 → ain X26 X25 → ain X26 (aint X24 X25)) → (∀X24 : set, asubq X24 a0 → (X24 = a0)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → ain X26 X24 → (¬ ain X26 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → asubq a1 (asing a0) → (∀X24 : set, ∀X25 : set, aal5 (aun X24 X25) → (aal3 X24 ∨ aal3 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X26 (aun X24 X25) → aal4 (asm (aun X24 X25) (asing X26)) → (aal2 X24 ∨ aal4 X25)) → (∀X24 : set, ∀X25 : set, aal5 X24 → (¬ aal3 (aint X24 X25)) → aal3 (asm X24 X25)) → (¬ (a0 = a2)) → ain a1 (asm (apow a2) (asing a2)) → (¬ asubq a2 (asing a1)) → (¬ ain (asm a3 (asing a1)) (asing (asing a1))) → (¬ ain (asm a3 (asing a1)) (apow a2)) → (∀X24 : set, ain X24 (asing a2) → (X24 = a2)) → (¬ ain (apow a2) a3) → adisjoint (asm (apow a2) a2) (apow (asing a2)) → asubq (asm (apow a2) a2) (asm (apow a2) (asing a1)) → (¬ ain (asm (apow a2) (apow (asing a2))) (asm (apow a2) (asing a1))) → ain (asing (asing a1)) (apow (asing (asing a1))) → ain a0 (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) → ain (asing a1) (aun (aun (asing a1) (asing (asing a1))) (apow (asing (asing a1)))) → ain a1 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → ain (asing a2) (apow (asing a2)) → ain a1 (aun (asing a1) (apow (asing a2))) → (¬ ain a3 (aun (apow a2) (asing (asing a2)))) → ain a0 (apow (asm a3 (asing a1))) → (¬ ain (asing (asing a1)) a4) → ain a3 (asm a4 a2) → ain a0 (aun (apow (asing a2)) (asing a3)) → (¬ ain (asing a1) (aun (asing (asing a2)) a4)) → ain a3 (aun (asing (asing a2)) a4) → ain (asm (apow a2) a2) (aun a3 (asing (asm (apow a2) a2))) → ain a0 (aun a2 (asing (apow a2))) → ain a0 (aun a3 (asing (apow a2))) → ain (apow a2) (aun a3 (asing (apow a2))) → ain a0 (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → ain (apow a2) (aun (apow (asing a2)) (asing (apow a2))) → (¬ ain (asm a3 (asing a1)) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → adisjoint a2 (aun (asing a3) (asing (apow a2))) → (¬ ain (asing a2) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → ((((X24 = a0) ∨ (X24 = asing a1)) ∨ (X24 = a2)) ∨ (X24 = asing (asing a1)))) → asubq (aun a3 (asing (asing a2))) (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → asubq (apow (asing (asing a1))) (aun (asing (asing (asing a1))) (aun a1 (asing (apow a2)))) → asubq (asm (apow (apow (asing a1))) (asing (apow (asing a1)))) (aun (asing (asing (asing a1))) (aun a2 (asing (apow a2)))) → asubq (asm (apow a2) (asing a2)) (aun (asm (apow a2) (asing a2)) (asing (asm (apow a2) (apow (asing a2))))) → (¬ asubq (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) (aun a4 (asing (apow a2)))) → (asm a2 (asing a0) = asing a1) → (asm a2 (asing a1) = a1) → asubq (asm (asm (apow a2) (asing a2)) (asing a0)) (aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing a0) = aun (asing a1) (asing (asing a1))) → (asm (asm (apow a2) (asing a2)) (asing (asing a1)) = a2) → asubq (asm (asm (apow a2) (apow (asing a1))) (asing a2)) (asing a1) → asubq (asm a3 (asing a1)) (asm (asm (apow a2) (asing a1)) (asing (asing a1))) → asubq (asm (asm (apow a2) (apow (asing a2))) (asing (asing a1))) (asm (apow a2) (apow (asing a1))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a2)) → ain X24 (aun (asing a1) (asing (asing a1)))) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = asing (asing a1))) → ain X24 (apow a2)) → asubq (apow a2) (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing (asing (asing a1)))) → (asm (asm a4 a2) (asing a2) = asing a3) → (∀X24 : set, ain X24 (asm a4 (asing a1)) → (¬ (X24 = a0)) → ain X24 (asm a4 a2)) → asubq (asm (asm a4 (asing a1)) (asing a3)) (asm a3 (asing a1)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → asubq (asing a2) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) → (aint (aun (apow a2) (asing (asing a2))) (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) = aun (asing a2) (apow (asing a2))) → (aint (aun (aun (asing a2) (apow (asing a2))) (asing (asm a3 (asing a1)))) (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → asubq (aint (aun (apow a2) (asing (asing a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (∀X24 : set, ain X24 a4 → (¬ ain X24 a2) → (¬ aal2 (asm (asm a4 a2) (asing X24)))) → (∀X24 : set, ain X24 (asm (apow a2) (asing a1)) → (¬ aal3 (asm (asm (apow a2) (asing a1)) (asing X24)))) → (¬ aal6 (aun (apow a2) (asing (asing a2)))) → (¬ aal6 (aun (apow a2) (asing (apow a2)))) → aal2 (asm (apow a2) (apow (asing a1))) → aal5 (aun (asing (asing a1)) a4) → aex2 (aun (asing (asm (apow a2) (apow (asing a2)))) (asing (apow a2))) → aex2 (asm (asm a4 (asing a2)) (asing a1)) → aex2 (asm (asm a4 (asing a1)) (asing a3)) → aex3 (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → aex5 (aun (apow a2) (asing (apow a2))) → (¬ aal4 (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing (asing a2)))) → (∀X24 : set, ain X24 a4 → (¬ ain X24 (asing a0)) → ain X24 (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) → ain X24 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1)))) → asubq (asm (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) (asing a0)) (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → aex5 (asm (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) (asing a1)) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → ain X24 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (¬ ain X24 (asing a1)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2))))) → asubq (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing (apow a2))) (aun a3 (asing (asing a2))) → (∀X24 : set, ain X24 (aun (apow (asing a2)) (asing (apow a2))) → (¬ aal5 (asm (aint (aun (asm (apow a2) (apow (asing a2))) (aun (apow (asing a2)) (asing (apow a2)))) (aun (asing (asing a2)) (aun a4 (asing (apow a2))))) (asing X24)))) → asubq (aun a4 (asing (apow a2))) (aun (asing (asing a1)) (aun a4 (asing (apow a2))))
In Proofgold the corresponding term root is 0e0b88... and proposition id is b34094...
Proof:
The rest of this subproof is missing.
∎
Theorem. (conj_AbstrHF_TMVadejEwVPWQeoChMX7BZdYC6FL1S2s21U)
(∀X24 : set, ∀X25 : set, (adisjoint X24 X25 ↔ (aint X24 X25 = a0))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (ain X26 (aint X24 X25) ↔ (ain X26 X24 ∧ ain X26 X25))) → (∀X24 : set, ∀X25 : set, ain X25 (apow X24) → asubq X25 X24) → (∀X24 : set, ∀X25 : set, asubq X25 (aun X24 X25)) → (∀X24 : set, ∀X25 : set, ∀X26 : set, (∀X27 : set, ain X27 X24 → ain X27 X25 → ain X27 X26) → asubq (aint X24 X25) X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, adisjoint X24 X25 → asubq X26 X25 → adisjoint X24 X26) → (∀X24 : set, ∀X25 : set, ∀X26 : set, ain X25 X24 → ain X26 X24 → (¬ asubq X26 X25) → aal2 X24) → (∀X24 : set, ∀X25 : set, ain X25 X24 → (¬ aal4 X24) → (¬ aal3 (asm X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, adisjoint X24 X25 → aal2 X25 → (∀X26 : set, ain X26 X25 → (¬ asubq (aun X24 X25) (aun X24 (asing X26))))) → (∀X24 : set, ∀X25 : set, (¬ aal4 X24) → (¬ aal5 (aun X24 (asing X25)))) → (∀X24 : set, ∀X25 : set, ∀X26 : set, asubq X26 (aun X24 X25) → (∀X27 : set, ain X27 X24 → (¬ ain X27 X26) → aal4 X26 → (aal3 X24 ∨ aal3 X25))) → (∀X24 : set, ain a0 X24 → asubq a1 X24) → (¬ ain a0 (asing a1)) → (¬ ain a1 (asing (asing a1))) → (∀X24 : set, ain X24 (apow (asing a1)) → ((X24 = a0) ∨ (X24 = asing a1))) → (¬ ain (asing a2) (apow a2)) → (¬ (a2 = asm (apow a2) a2)) → (¬ asubq (apow a2) (asing a2)) → ain (asing (asing a1)) (asm (apow (aun (asing a1) (asing (asing a1)))) (asing (aun (asing a1) (asing (asing a1))))) → ain (aun (asing a1) (asing (asing a1))) (asing (aun (asing a1) (asing (asing a1)))) → (¬ ain (asing a2) a4) → (¬ ain a1 (aun (asing a2) (apow (asing a2)))) → (¬ ain (asing a1) (aun (asing a2) (apow (asing a2)))) → ain a2 (aun (asing a2) (apow (asing a2))) → ain a3 (asm a4 (asing a1)) → ain a1 (asm (aun (asing (asing a2)) a4) (asm (apow a2) (asing a1))) → (¬ ain (asm (apow a2) a2) (aun (asing (asing a2)) a4)) → ain (asm (apow a2) (apow (asing a2))) (asing (asm (apow a2) (apow (asing a2)))) → ain (asm (apow a2) (apow (asing a2))) (aun (asm (apow a2) (asing a1)) (asing (asm (apow a2) (apow (asing a2))))) → ain a1 (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → ain a1 (aun (asing (asing a1)) (aun a4 (asing (apow a2)))) → ain a2 (aun (asing (asing a2)) (aun a4 (asing (apow a2)))) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (((X24 = a1) ∨ (X24 = asing a1)) ∨ (X24 = a2))) → (∀X24 : set, ain X24 (asm a4 (apow (asing a2))) → (((X24 = a1) ∨ (X24 = a2)) ∨ (X24 = a3))) → asubq a3 (aun (asm (apow a2) (apow (asing a1))) (apow (asing (asing a1)))) → (¬ asubq (aun (apow a2) (asing (apow a2))) (apow a2)) → (∀X24 : set, ain X24 (asm (apow a2) (asing a2)) → (¬ (X24 = a0)) → ain X24 (aun (asing a1) (asing (asing a1)))) → asubq a2 (asm (asm (apow a2) (asing a2)) (asing (asing a1))) → asubq (asm (asm a3 (asing a1)) (asing a0)) (asing a2) → (∀X24 : set, ain X24 (asm (apow a2) (apow (asing a2))) → (¬ (X24 = a1)) → ain X24 (asm (apow a2) a2)) → (∀X24 : set, ain X24 (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) → (¬ (X24 = a1)) → ain X24 (aun (asm (apow a2) a2) (apow (asing (asing a1))))) → asubq (asm (aun (asm (apow a2) (apow (asing a2))) (apow (asing (asing a1)))) (asing a1)) (aun (asm (apow a2) a2) (apow (asing (asing a1)))) → asubq (asing a3) (asm (asm a4 a2) (asing a2)) → asubq (asm (asm a4 (apow (asing a2))) (asing a2)) (aun (asing a1) (asing a3)) → asubq (asm (apow a2) (apow (asing a1))) (asm (asm a4 (apow (asing a2))) (asing a3)) → (aint (aun (apow a2) (asing (asing a2))) (aun a4 (asing (asm (apow a2) a2))) = a3) → (aint (aun (apow a2) (asing (asing a2))) (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) = aun (asing a2) (apow (asing a2))) → (∀X24 : set, ain X24 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → ain X24 (aun (asing a2) (aun (apow (asing a2)) (asing (apow a2)))) → ain X24 (aun (asing a2) (apow (asing a2)))) → asubq (aint (aun (apow a2) (asing (apow a2))) (asm a4 (asing a1))) (asm a3 (asing a1)) → (¬ aal2 (asing a1)) → (¬ aal3 (asm (apow a2) (apow (asing a1)))) → (¬ aal7 (aun (asing (asing a1)) (aun (asing (asing a2)) a4))) → aal2 (asm a3 (asing a1)) → aal5 (aun (asing (asing (asing a1))) a4) → aal6 (aun (asing (asing a1)) (aun (asing (asing a2)) a4)) → aex3 (asm (apow a2) (asing a2)) → aex3 (asm (apow a2) (asing (asing a1))) → aex4 (aint (aun (aun (asing a1) (asing (asing a1))) (aun (apow (asing a2)) (asing a3))) (aun (asing (asing a2)) a4)) → asubq (aun (apow (asing a2)) (asing (apow a2))) (aun (asing a1) (aun (apow (asing a2)) (asing (apow a2)))) → (∀X24 : set, ∀X25 : set, (ain X25 (apow X24) ↔ asubq X25 X24))
In Proofgold the corresponding term root is 82703e... and proposition id is 10a741...
Proof:
The rest of this subproof is missing.
∎
End of Section AbstrHF