Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Assume H0: x0 ⊆ x1.
Assume H1:
∀ x3 . x3 ∈ x1 ⟶ nIn x3 x0 ⟶ 0 ∈ x2 x3.
Let x3 of type ι be given.
Assume H2:
x3 ∈ Pi x0 (λ x4 . x2 x4).
Apply PiE with
x0,
x2,
x3,
x3 ∈ Pi x1 (λ x4 . x2 x4) leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3:
∀ x4 . x4 ∈ x3 ⟶ and (pair_p x4) (ap x4 0 ∈ x0).
Assume H4:
∀ x4 . x4 ∈ x0 ⟶ ap x3 x4 ∈ x2 x4.
Apply PiI with
x1,
x2,
x3 leaving 2 subgoals.
Let x4 of type ι be given.
Assume H5: x4 ∈ x3.
Apply H3 with
x4,
and (pair_p x4) (ap x4 0 ∈ x1) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply andI with
pair_p x4,
ap x4 0 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
ap x4 0.
The subproof is completed by applying H7.
Let x4 of type ι be given.
Assume H5: x4 ∈ x1.
Apply xm with
x4 ∈ x0,
ap x3 x4 ∈ x2 x4 leaving 2 subgoals.
Assume H6: x4 ∈ x0.
Apply H4 with
x4.
The subproof is completed by applying H6.
Apply Pi_eta with
x0,
x2,
x3,
λ x5 x6 . ap x5 x4 = 0 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply beta0 with
x0,
ap x3,
x4.
The subproof is completed by applying H6.
Apply L7 with
λ x5 x6 . x6 ∈ x2 x4.
Apply H1 with
x4 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.