Let x0 of type ι be given.
Apply ZF_closed_E with
x0,
∀ x1 . x1 ∈ x0 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x0) ⟶ Pi x1 (λ x3 . x2 x3) ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Assume H5: x1 ∈ x0.
Let x2 of type ι → ι be given.
Assume H6: ∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x0.
The subproof is completed by applying Sep_In_Power with
prim4 (lam x1 (λ x3 . prim3 (x2 x3))),
λ x3 . ∀ x4 . x4 ∈ x1 ⟶ ap x3 x4 ∈ x2 x4.
Apply H3 with
prim4 (lam x1 (λ x3 . prim3 (x2 x3))).
Apply H3 with
lam x1 (λ x3 . prim3 (x2 x3)).
Apply unknownprop_d9ae82204b18e6cf15c85d865639887282bf3ebbe7f609859927820b6a09adb1 with
x0,
x1,
λ x3 . prim3 (x2 x3) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
Let x3 of type ι be given.
Assume H8: x3 ∈ x1.
Apply H2 with
x2 x3.
Apply H6 with
x3.
The subproof is completed by applying H8.
Apply H0 with
prim4 (prim4 (lam x1 (λ x3 . prim3 (x2 x3)))),
Pi x1 (λ x3 . x2 x3) leaving 2 subgoals.
The subproof is completed by applying L8.
The subproof is completed by applying L7.