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 . and (x1 x3) (x3 = x0 ⟶ ∀ x4 : ο . x4)) x2 leaving 2 subgoals.
Assume H1: x1 x2.
Apply andI with
x1 x2,
x2 = x0 ⟶ ∀ x3 : ο . x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2: x2 = x0.
Apply In_irref with
x0.
Apply H2 with
λ x3 x4 . x3 ∈ x0.
The subproof is completed by applying H0.
Assume H1:
(λ x3 . and (x1 x3) (x3 = x0 ⟶ ∀ x4 : ο . x4)) x2.
Apply H1 with
x1 x2.
Assume H2: x1 x2.
Assume H3: x2 = x0 ⟶ ∀ x3 : ο . x3.
The subproof is completed by applying H2.