Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H0: x2 ∈ x0.
Apply set_ext with
ap (lam x0 (λ x3 . x1 x3)) x2,
x1 x2 leaving 2 subgoals.
Let x3 of type ι be given.
Assume H1:
x3 ∈ ap (lam x0 (λ x4 . x1 x4)) x2.
Claim L2:
setsum x2 x3 ∈ lam x0 (λ x4 . x1 x4)Apply apE with
lam x0 (λ x4 . x1 x4),
x2,
x3.
The subproof is completed by applying H1.
Apply pair_Sigma_E1 with
x0,
x1,
x2,
x3.
The subproof is completed by applying L2.
Let x3 of type ι be given.
Assume H1: x3 ∈ x1 x2.
Apply apI with
lam x0 (λ x4 . x1 x4),
x2,
x3.
Apply lamI with
x0,
λ x4 . x1 x4,
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.