Let x0 of type ι be given.
Let x1 of type ο be given.
Assume H0:
∀ x2 : ι → ι . bij (setexp x0 0) 1 x2 ⟶ x1.
Apply H0 with
λ x2 . 0.
Apply bijI with
setexp x0 0,
1,
λ x2 . 0 leaving 3 subgoals.
Let x2 of type ι be given.
The subproof is completed by applying In_0_1.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H3: (λ x4 . 0) x2 = (λ x4 . 0) x3.
Apply Pi_ext with
0,
λ x4 . x0,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x4 of type ι be given.
Assume H4: x4 ∈ 0.
Apply FalseE with
ap x2 x4 = ap x3 x4.
Apply EmptyE with
x4.
The subproof is completed by applying H4.
Let x2 of type ι be given.
Assume H1: x2 ∈ 1.
Let x3 of type ο be given.
Assume H2:
∀ x4 . and (x4 ∈ setexp x0 0) ((λ x5 . 0) x4 = x2) ⟶ x3.
Apply H2 with
0.
Apply andI with
0 ∈ setexp x0 0,
0 = x2 leaving 2 subgoals.
Apply PiI with
0,
λ x4 . x0,
0 leaving 2 subgoals.
Let x4 of type ι be given.
Assume H3: x4 ∈ 0.
Apply FalseE with
and (pair_p x4) (ap x4 0 ∈ 0).
Apply EmptyE with
x4.
The subproof is completed by applying H3.
Let x4 of type ι be given.
Assume H3: x4 ∈ 0.
Apply FalseE with
ap 0 x4 ∈ x0.
Apply EmptyE with
x4.
The subproof is completed by applying H3.
Apply cases_1 with
x2,
λ x4 . 0 = x4 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι → ι → ο be given.
Assume H3: x4 0 0.
The subproof is completed by applying H3.