Primitive. The name Eps_i is a term of type (set → prop) → set.
L2
Axiom. (Eps_i_ax) We take the following as an axiom:
∀P : set → prop, ∀x : set, P x → P (Eps_i P)
In Proofgold the corresponding term root is c3f0de... and proposition id is 756d6e...
L4
Definition. We define True to be ∀p : prop, p → p of type prop.
In Proofgold the corresponding term root is f81b39... and object id is 94e6a7...
L6
Definition. We define False to be ∀p : prop, p of type prop.
In Proofgold the corresponding term root is 1db705... and object id is 266ad5...
L7
Definition. We define not to be λA : prop ⇒ A → False of type prop → prop.
In Proofgold the corresponding term root is f30435... and object id is 0dc3c7...
Notation. We use ¬ as a prefix operator with priority 700 corresponding to applying term not.
L12
Definition. We define and to be λA B : prop ⇒ ∀p : prop, (A → B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 2ba7d0... and object id is 37da83...
Notation. We use ∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term and.
L17
Definition. We define or to be λA B : prop ⇒ ∀p : prop, (A → p) → (B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 957746... and object id is 86ae64...
Notation. We use ∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term or.
L22
Definition. We define iff to be λA B : prop ⇒ and (A → B) (B → A) of type prop → prop → prop.
In Proofgold the corresponding term root is 98aaaf... and object id is 6588e5...
Notation. We use ↔ as an infix operator with priority 805 and no associativity corresponding to applying term iff.
Beginning of Section Eq
L29
Variable A : SType
L30
Definition. We define eq to be λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type A → A → prop.
L31
Definition. We define neq to be λx y : A ⇒ ¬ eq x y of type A → A → prop.
End of Section Eq
Notation. We use = as an infix operator with priority 502 and no associativity corresponding to applying term eq.
Notation. We use ≠ as an infix operator with priority 502 and no associativity corresponding to applying term neq.
L37
Theorem. (t3_xregular)
∀k3_tarski : set → set, ∀esk6_2 : set → set → set, ∀v1_xboole_0 : set → prop, ∀esk10_1 : set → set, ∀esk11_2 : set → set → set, ∀r1_xboole_0 : set → set → prop, ∀esk12_1 : set → set, ∀esk14_2 : set → set → set, ∀esk1_0 : set, ∀esk2_1 : set → set, ∀esk13_1 : set → set, ∀esk9_0 : set, ∀esk8_0 : set, ∀k1_xboole_0 : set, ∀esk3_1 : set → set, ∀esk7_2 : set → set → set, ∀esk5_3 : set → set → set → set, ∀r2_hidden : set → set → prop, ∀esk4_3 : set → set → set → set, ∀k2_xboole_0 : set → set → set, (∀X1 X3 X2, (X3 = (k2_xboole_0 X1 X2) → False) → r2_hidden (esk4_3 X1 X2 X3) X3 → r2_hidden (esk4_3 X1 X2 X3) X1 → False) → (∀X2 X3 X1, (X3 = (k2_xboole_0 X1 X2) → False) → r2_hidden (esk4_3 X1 X2 X3) X3 → r2_hidden (esk4_3 X1 X2 X3) X2 → False) → (∀X1 X3 X2, (X3 = (k2_xboole_0 X1 X2) → False) → (r2_hidden (esk4_3 X1 X2 X3) X3 → False) → (r2_hidden (esk4_3 X1 X2 X3) X2 → False) → (r2_hidden (esk4_3 X1 X2 X3) X1 → False) → False) → (∀X3 X2 X1, (X2 = (k3_tarski X1) → False) → r2_hidden X3 X1 → r2_hidden (esk6_2 X1 X2) X3 → r2_hidden (esk6_2 X1 X2) X2 → False) → (∀X1 X2 X3, (r2_hidden (esk5_3 X1 X2 X3) X1 → False) → X2 = (k3_tarski X1) → r2_hidden X3 X2 → False) → (∀X2 X3 X1, (r2_hidden X1 (esk5_3 X2 X3 X1) → False) → X3 = (k3_tarski X2) → r2_hidden X1 X3 → False) → (∀X1 X2, (X2 = (k3_tarski X1) → False) → (r2_hidden (esk6_2 X1 X2) X2 → False) → (r2_hidden (esk6_2 X1 X2) (esk7_2 X1 X2) → False) → False) → (∀X2 X1, (v1_xboole_0 X1 → False) → r2_hidden X2 (esk10_1 X1) → r1_xboole_0 (esk11_2 X1 X2) X1 → False) → (∀X3 X1 X2, (v1_xboole_0 X2 → False) → (r1_xboole_0 X3 X2 → False) → (r2_hidden X1 (esk10_1 X2) → False) → r2_hidden X3 X1 → r2_hidden X1 (k3_tarski X2) → False) → (∀X1 X2, (v1_xboole_0 X2 → False) → (r1_xboole_0 X1 X2 → False) → (r2_hidden X1 (esk12_1 X2) → False) → r2_hidden X1 (k3_tarski (k3_tarski X2)) → False) → (∀X1 X2, (X2 = (k3_tarski X1) → False) → (r2_hidden (esk6_2 X1 X2) X2 → False) → (r2_hidden (esk7_2 X1 X2) X1 → False) → False) → (∀X3 X2 X1, (r1_xboole_0 X1 (k2_xboole_0 X2 X3) → False) → r1_xboole_0 X1 X3 → r1_xboole_0 X1 X2 → False) → (∀X3 X1 X2, (r1_xboole_0 X1 X2 → False) → r1_xboole_0 X1 (k2_xboole_0 X2 X3) → False) → (∀X3 X1 X2, (r1_xboole_0 X1 X2 → False) → r1_xboole_0 X1 (k2_xboole_0 X3 X2) → False) → (∀X2 X1, (v1_xboole_0 X1 → False) → (r2_hidden (esk11_2 X1 X2) X2 → False) → r2_hidden X2 (esk10_1 X1) → False) → (∀X3 X2 X1 X4, (r2_hidden X1 X4 → False) → (r2_hidden X1 X3 → False) → X2 = (k2_xboole_0 X3 X4) → r2_hidden X1 X2 → False) → (∀X1 X2, (v1_xboole_0 X2 → False) → (r2_hidden X1 (k3_tarski (k3_tarski X2)) → False) → r2_hidden X1 (esk12_1 X2) → False) → (∀X3 X2 X1 X4, (r2_hidden X1 X4 → False) → X4 = (k3_tarski X3) → r2_hidden X2 X3 → r2_hidden X1 X2 → False) → (∀X2 X1 X3, r2_hidden X1 X3 → r2_hidden X1 X2 → r1_xboole_0 X2 X3 → False) → (∀X2 X4 X1 X3, (r2_hidden X1 X3 → False) → X3 = (k2_xboole_0 X2 X4) → r2_hidden X1 X2 → False) → (∀X2 X4 X1 X3, (r2_hidden X1 X3 → False) → X3 = (k2_xboole_0 X4 X2) → r2_hidden X1 X2 → False) → (∀X1 X2, (v1_xboole_0 X2 → False) → r1_xboole_0 X1 X2 → r2_hidden X1 (esk12_1 X2) → False) → (∀X1 X2, (v1_xboole_0 X2 → False) → (r2_hidden X1 (k3_tarski X2) → False) → r2_hidden X1 (esk10_1 X2) → False) → (∀X1 X2, (r1_xboole_0 X1 X2 → False) → (r2_hidden (esk14_2 X1 X2) X1 → False) → False) → (∀X2 X1, (r1_xboole_0 X1 X2 → False) → (r2_hidden (esk14_2 X1 X2) X2 → False) → False) → (∀X2 X1, (v1_xboole_0 X1 → False) → v1_xboole_0 (k2_xboole_0 X1 X2) → False) → (∀X2 X1, (v1_xboole_0 X1 → False) → v1_xboole_0 (k2_xboole_0 X2 X1) → False) → (∀X1, r2_hidden X1 esk1_0 → r1_xboole_0 (esk2_1 X1) esk1_0 → False) → (∀X1, (r2_hidden (esk2_1 X1) (esk3_1 X1) → False) → r2_hidden X1 esk1_0 → False) → (∀X2 X1, r2_hidden X2 X1 → r2_hidden X1 X2 → False) → (∀X1, (r2_hidden (esk3_1 X1) X1 → False) → r2_hidden X1 esk1_0 → False) → (∀X2 X1, (r1_xboole_0 X2 X1 → False) → r1_xboole_0 X1 X2 → False) → (∀X1 X2, v1_xboole_0 X2 → r2_hidden X1 X2 → False) → (∀X1, (v1_xboole_0 X1 → False) → (r1_xboole_0 (esk13_1 X1) X1 → False) → False) → (∀X1, (v1_xboole_0 X1 → False) → (r2_hidden (esk13_1 X1) X1 → False) → False) → (∀X2 X1, (X1 = X2 → False) → v1_xboole_0 X2 → v1_xboole_0 X1 → False) → (∀X1, (X1 = k1_xboole_0 → False) → v1_xboole_0 X1 → False) → (v1_xboole_0 esk9_0 → False) → (v1_xboole_0 esk1_0 → False) → (∀X2 X3 X1, ((k2_xboole_0 (k2_xboole_0 X1 X2) X3) = (k2_xboole_0 X1 (k2_xboole_0 X2 X3)) → False) → False) → (∀X2 X1, ((k2_xboole_0 X1 X2) = (k2_xboole_0 X2 X1) → False) → False) → (∀X1, ((k2_xboole_0 X1 X1) = X1 → False) → False) → (∀X1, ((k2_xboole_0 X1 k1_xboole_0) = X1 → False) → False) → ((v1_xboole_0 esk8_0 → False) → False) → ((v1_xboole_0 k1_xboole_0 → False) → False) → False
In Proofgold the corresponding term root is 0d75c7... and proposition id is 08f54a...
Proof:
Proof not loaded.