Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H0:
x2 ∈ Pi x0 (λ x3 . x1 x3).
Apply SepE with
prim4 (lam x0 (λ x3 . prim3 (x1 x3))),
λ x3 . ∀ x4 . x4 ∈ x0 ⟶ ap x3 x4 ∈ x1 x4,
x2,
and (∀ x3 . x3 ∈ x2 ⟶ and (pair_p x3) (ap x3 0 ∈ x0)) (∀ x3 . x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H2:
∀ x3 . x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3.
Apply andI with
∀ x3 . x3 ∈ x2 ⟶ and (pair_p x3) (ap x3 0 ∈ x0),
∀ x3 . x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3 leaving 2 subgoals.
Let x3 of type ι be given.
Assume H3: x3 ∈ x2.
Claim L4:
x2 ⊆ lam x0 (λ x4 . prim3 (x1 x4))Apply PowerE with
lam x0 (λ x4 . prim3 (x1 x4)),
x2.
The subproof is completed by applying H1.
Claim L5:
x3 ∈ lam x0 (λ x4 . prim3 (x1 x4))Apply L4 with
x3.
The subproof is completed by applying H3.
Apply andI with
pair_p x3,
ap x3 0 ∈ x0 leaving 2 subgoals.
Apply proj_Sigma_eta with
x0,
λ x4 . prim3 (x1 x4),
x3.
The subproof is completed by applying L5.
Apply L6 with
λ x4 x5 . pair_p x4.
The subproof is completed by applying pair_p_I with
proj0 x3,
proj1 x3.
Apply ap0_Sigma with
x0,
λ x4 . prim3 (x1 x4),
x3.
The subproof is completed by applying L5.
The subproof is completed by applying H2.