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 → ο be given.
Assume H2:
∀ x4 x5 . x1 x4 ⟶ x1 x5 ⟶ x2 = pair_tag x0 x4 x5 ⟶ x3 (pair_tag x0 x4 x5).
Apply H1 with
x3 x2.
Let x4 of type ι be given.
Assume H3:
(λ x5 . and (x1 x5) (∃ x6 . and (x1 x6) (x2 = pair_tag x0 x5 x6))) x4.
Apply H3 with
x3 x2.
Assume H4: x1 x4.
Assume H5:
∃ x5 . and (x1 x5) (x2 = pair_tag x0 x4 x5).
Apply H5 with
x3 x2.
Let x5 of type ι be given.
Assume H6:
(λ x6 . and (x1 x6) (x2 = pair_tag x0 x4 x6)) x5.
Apply H6 with
x3 x2.
Assume H7: x1 x5.
Apply H8 with
λ x6 x7 . x3 x7.
Apply H2 with
x4,
x5 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
The subproof is completed by applying H8.