Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Let x2 of type ι be given.
Assume H0: x2 ∈ x0.
Apply iffI with
x1 x2,
(λ x3 . or (x1 x3) (x3 = x0)) x2 leaving 2 subgoals.
Assume H1: x1 x2.
Apply orIL with
x1 x2,
x2 = x0.
The subproof is completed by applying H1.
Assume H1:
(λ x3 . or (x1 x3) (x3 = x0)) x2.
Apply H1 with
x1 x2 leaving 2 subgoals.
Assume H2: x1 x2.
The subproof is completed by applying H2.
Assume H2: x2 = x0.
Apply FalseE with
x1 x2.
Apply In_irref with
x0.
Apply H2 with
λ x3 x4 . x3 ∈ x0.
The subproof is completed by applying H0.