Let x0 of type ι be given.
Apply In_ind with
λ x1 . x1 ∈ x0 ⟶ Inj1 x1 ∈ x0.
Let x1 of type ι be given.
Assume H2:
∀ x2 . x2 ∈ x1 ⟶ x2 ∈ x0 ⟶ Inj1 x2 ∈ x0.
Assume H3: x1 ∈ x0.
Apply Inj1_eq with
x1,
λ x2 x3 . x3 ∈ x0.
Apply ZF_binunion_closed with
x0,
Sing 0,
{Inj1 x2|x2 ∈ x1} leaving 3 subgoals.
The subproof is completed by applying H1.
Apply ZF_Sing_closed with
x0,
0 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H0 with
prim4 x1,
0 leaving 2 subgoals.
Apply ZF_Power_closed with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying Empty_In_Power with x1.
Apply ZF_Repl_closed with
x0,
x1,
λ x2 . Inj1 x2 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Assume H4: x2 ∈ x1.
Apply H2 with
x2 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.