Let x0 of type ι be given.
Apply set_ext with
binunion 0 x0,
x0 leaving 2 subgoals.
Let x1 of type ι be given.
Apply binunionE with
0,
x0,
x1,
x1 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H1: x1 ∈ 0.
Apply FalseE with
x1 ∈ x0.
Apply EmptyE with
x1.
The subproof is completed by applying H1.
Assume H1: x1 ∈ x0.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Assume H0: x1 ∈ x0.
Apply binunionI2 with
0,
x0,
x1.
The subproof is completed by applying H0.