Let x0 of type ι → ι be given.
Apply H0 with
False.
Apply PowerI with
omega,
{x1 ∈ omega|nIn x1 (x0 x1)}.
Let x1 of type ι be given.
Assume H3:
x1 ∈ {x2 ∈ omega|nIn x2 (x0 x2)}.
Apply SepE1 with
omega,
λ x2 . nIn x2 (x0 x2),
x1.
The subproof is completed by applying H3.
Apply H2 with
{x1 ∈ omega|nIn x1 (x0 x1)},
False leaving 2 subgoals.
The subproof is completed by applying L3.
Let x1 of type ι be given.
Apply H4 with
False.
Assume H6:
x0 x1 = {x2 ∈ omega|nIn x2 (x0 x2)}.
Assume H7:
x1 ∈ {x2 ∈ omega|nIn x2 (x0 x2)}.
Apply SepE2 with
omega,
λ x2 . nIn x2 (x0 x2),
x1 leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H6 with
λ x2 x3 . x1 ∈ x3.
The subproof is completed by applying H7.
Apply L7.
Apply SepI with
omega,
λ x2 . nIn x2 (x0 x2),
x1 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H6 with
λ x2 x3 . nIn x1 x3.
The subproof is completed by applying L7.