Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Assume H0:
∀ x2 . x1 x2 ⟶ ∀ x3 . x3 ∈ x2 ⟶ nIn x0 x3.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H1: x1 x2.
Assume H2: x1 x3.
Let x4 of type ο be given.
Apply H3 with
x2.
Apply andI with
x1 x2,
∃ x5 . and (x1 x5) (pair_tag x0 x2 x3 = pair_tag x0 x2 x5) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x5 of type ο be given.
Apply H4 with
x3.
Apply andI with
x1 x3,
pair_tag x0 x2 x3 = pair_tag x0 x2 x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x6 of type ι → ι → ο be given.
The subproof is completed by applying H5.