Let x0 of type ι be given.
Let x1 of type ι be given.
Apply unknownprop_4ad571f4a0c6aaf98e7cea1d7a5e094b177421e3e4501781303e29c4a79ec502 with
λ x2 x3 : ι → ο . x3 x0 ⟶ x3 x1 ⟶ (∀ x4 x5 . iff (In (lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) x0) (In (lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) x1)) ⟶ x0 = x1.
Assume H0:
(λ x2 . ∀ x3 . In x3 x2 ⟶ ∃ x4 x5 . x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) x0.
Assume H1:
(λ x2 . ∀ x3 . In x3 x2 ⟶ ∃ x4 x5 . x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) x1.
Assume H2:
∀ x2 x3 . iff (In (lam 2 (λ x4 . If_i (x4 = 0) x2 x3)) x0) (In (lam 2 (λ x4 . If_i (x4 = 0) x2 x3)) x1).
Apply unknownprop_219a5692ece616b4a88502d80a85b644180cde982b21251f92a23d11d1a5d022 with
x0,
x1 leaving 2 subgoals.
Let x2 of type ι be given.
Apply H0 with
x2,
In x2 x1 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4:
(λ x4 . ∃ x5 . x2 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) x3.
Apply H4 with
In x2 x1.
Let x4 of type ι be given.
Assume H5:
x2 = lam 2 (λ x5 . If_i (x5 = 0) x3 x4).
Apply H5 with
λ x5 x6 . In x6 x1.
Apply unknownprop_6e9d790c24657bc527a0f62de036403ca00386b366ddc02915f8c3a4de529eee with
In (lam 2 (λ x5 . If_i (x5 = 0) x3 x4)) x0,
In (lam 2 (λ x5 . If_i (x5 = 0) x3 x4)) x1,
In (lam 2 (λ x5 . If_i (x5 = 0) x3 x4)) x1 leaving 3 subgoals.
The subproof is completed by applying H2 with x3, x4.
Assume H6:
In (lam 2 (λ x5 . If_i (x5 = 0) x3 x4)) x0.
Assume H7:
In (lam 2 (λ x5 . If_i (x5 = 0) x3 x4)) x1.
The subproof is completed by applying H7.
Assume H6:
not (In (lam 2 (λ x5 . If_i (x5 = 0) x3 x4)) x0).
Assume H7:
not (In (lam 2 (λ x5 . If_i (x5 = 0) x3 x4)) x1).
Apply FalseE with
In (lam 2 (λ x5 . If_i (x5 = 0) x3 x4)) x1.
Apply notE with
In (lam 2 (λ x5 . If_i (x5 = 0) x3 x4)) x0 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H5 with
λ x5 x6 . In x5 x0.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Apply H1 with
x2,
In x2 x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4: ....