Primitive. The name
Eps_i is a term of type
(set → prop) → set.
Axiom. (
Eps_i_ax) We take the following as an axiom:
∀P : set → prop, ∀x : set, P x → P (Eps_i P)
Definition. We define
True to be
∀p : prop, p → p of type
prop.
Definition. We define
False to be
∀p : prop, p of type
prop.
Definition. We define
not to be
λA : prop ⇒ A → False of type
prop → prop.
Notation. We use
¬ as a prefix operator with priority 700 corresponding to applying term
not.
Definition. We define
and to be
λA B : prop ⇒ ∀p : prop, (A → B → p) → p of type
prop → prop → prop.
Notation. We use
∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term
and.
Definition. We define
or to be
λA B : prop ⇒ ∀p : prop, (A → p) → (B → p) → p of type
prop → prop → prop.
Notation. We use
∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term
or.
Definition. We define
iff to be
λA B : prop ⇒ and (A → B) (B → A) of type
prop → prop → prop.
Notation. We use
↔ as an infix operator with priority 805 and no associativity corresponding to applying term
iff.
Beginning of Section Eq
Variable A : SType
Definition. We define
eq to be
λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type
A → A → prop.
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.
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
Proof:The rest of the proof is missing.