Let x0 of type ι be given.
Let x1 of type ι be given.
Apply Inj1_eq with
x0,
λ x2 x3 . x1 ∈ x3 ⟶ or (x1 = 0) (∃ x4 . and (x4 ∈ x0) (x1 = Inj1 x4)).
Apply binunionE with
Sing 0,
{Inj1 x2|x2 ∈ x0},
x1,
or (x1 = 0) (∃ x2 . and (x2 ∈ x0) (x1 = Inj1 x2)) leaving 3 subgoals.
The subproof is completed by applying H0.
Apply orIL with
x1 = 0,
∃ x2 . and (x2 ∈ x0) (x1 = Inj1 x2).
Apply SingE with
0,
x1.
The subproof is completed by applying H1.
Assume H1:
x1 ∈ {Inj1 x2|x2 ∈ x0}.
Apply orIR with
x1 = 0,
∃ x2 . and (x2 ∈ x0) (x1 = Inj1 x2).
Apply ReplE with
x0,
Inj1,
x1.
The subproof is completed by applying H1.