Let x0 of type ι be given.
Let x1 of type ι be given.
Apply binunionE with
x0,
Sing x0,
x1,
or (x1 ∈ x0) (x1 = x0) leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H1: x1 ∈ x0.
Apply orIL with
x1 ∈ x0,
x1 = x0.
The subproof is completed by applying H1.
Apply orIR with
x1 ∈ x0,
x1 = x0.
Apply SingE with
x0,
x1.
The subproof is completed by applying H1.