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) ⟶ lam 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.
Apply Union_Repl_famunion_closed with
x0 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Apply L7 with
x1,
λ x3 . {setsum x3 x4|x4 ∈ x2 x3} leaving 2 subgoals.
The subproof is completed by applying H5.
Let x3 of type ι be given.
Assume H8: x3 ∈ x1.
Apply H4 with
x2 x3,
λ x4 . setsum x3 x4 leaving 2 subgoals.
Apply H6 with
x3.
The subproof is completed by applying H8.
Let x4 of type ι be given.
Assume H9: x4 ∈ x2 x3.
Apply unknownprop_527cc5ff4380236ca54e2a0208bde8411cd2e1124e14f7764871099a03bebf91 with
x0,
x3,
x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply H0 with
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
Apply H0 with
x2 x3,
x4 leaving 2 subgoals.
Apply H6 with
x3.
The subproof is completed by applying H8.
The subproof is completed by applying H9.