Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H2: x1 ∈ x0.
Let x2 of type ι be given.
Assume H3: x2 ∈ x0.
Apply ZF_binunion_closed with
x0,
{Inj0 x3|x3 ∈ x1},
{Inj1 x3|x3 ∈ x2} leaving 3 subgoals.
The subproof is completed by applying H1.
Apply ZF_Repl_closed with
x0,
x1,
Inj0 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x3 of type ι be given.
Assume H4: x3 ∈ x1.
Apply unknownprop_219267f188d024efd66eafc845ecebe18fbe0d2f7334f1cc009aec407d26a1f3 with
x0,
x3 leaving 3 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 H2.
The subproof is completed by applying H4.
Apply ZF_Repl_closed with
x0,
x2,
Inj1 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4: x3 ∈ x2.
Apply unknownprop_6171474f197c8259ed73c167f3350e99942e9d4302f7776e81339ae25ce09a62 with
x0,
x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply H0 with
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.