Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H2:
∀ x2 x3 . iff (lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ x0) (lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ x1).
Apply set_ext with
x0,
x1 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H3: x2 ∈ x0.
Apply H0 with
x2,
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
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 . x6 ∈ x1.
Apply H2 with
x3,
x4,
lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x1.
Assume H6:
lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x0 ⟶ lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x1.
Assume H7:
lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x1 ⟶ lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x0.
Apply H6.
Apply H5 with
λ x5 x6 . x5 ∈ x0.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Assume H3: x2 ∈ x1.
Apply H1 with
x2,
x2 ∈ x0 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
x2 ∈ x0.
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 . x6 ∈ x0.
Apply H2 with
x3,
x4,
lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x0.
Assume H6:
lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x0 ⟶ lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x1.
Assume H7:
lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x1 ⟶ lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ x0.
Apply H7.
Apply H5 with
λ x5 x6 . x5 ∈ x1.
The subproof is completed by applying H3.