Let x0 of type ι be given.
Apply ZF_closed_E with
x0,
V_closed x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply Union_Repl_famunion_closed with
x0 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Claim L6:
∀ x1 . (∀ x2 . x2 ∈ x1 ⟶ x2 ∈ x0 ⟶ V_ x2 ∈ x0) ⟶ x1 ∈ x0 ⟶ V_ x1 ∈ x0Let x1 of type ι be given.
Assume H6:
∀ x2 . x2 ∈ x1 ⟶ x2 ∈ x0 ⟶ V_ x2 ∈ x0.
Assume H7: x1 ∈ x0.
Apply V_eq with
x1,
λ x2 x3 . x3 ∈ x0.
Apply L5 with
x1,
λ x2 . prim4 (V_ x2) leaving 2 subgoals.
The subproof is completed by applying H7.
Let x2 of type ι be given.
Assume H8: x2 ∈ x1.
Apply H3 with
V_ x2.
Claim L9: x2 ∈ x0
Apply H0 with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply H6 with
x2 leaving 2 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying L9.
Apply In_ind with
λ x1 . x1 ∈ x0 ⟶ V_ x1 ∈ x0.
The subproof is completed by applying L6.