Let x0 of type ι be given.
Apply Unj_eq with
Inj0 x0,
λ x1 x2 . x2 = x0.
Apply set_ext with
{Unj x1|x1 ∈ setminus (Inj0 x0) (Sing 0)},
x0 leaving 2 subgoals.
Let x1 of type ι be given.
Apply ReplE_impred with
setminus (Inj0 x0) (Sing 0),
Unj,
x1,
x1 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x2 of type ι be given.
Apply setminusE with
Inj0 x0,
Sing 0,
x2,
x1 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H3:
x2 ∈ {Inj1 x3|x3 ∈ x0}.
Apply ReplE_impred with
x0,
Inj1,
x2,
x1 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H5: x3 ∈ x0.
Claim L7: x1 = x3
Apply H2 with
λ x4 x5 . x5 = x3.
Apply H6 with
λ x4 x5 . Unj x5 = x3.
The subproof is completed by applying Unj_Inj1_eq with x3.
Apply L7 with
λ x4 x5 . x5 ∈ x0.
The subproof is completed by applying H5.
Let x1 of type ι be given.
Assume H0: x1 ∈ x0.
Apply Unj_Inj1_eq with
x1,
λ x2 x3 . x2 ∈ {Unj x4|x4 ∈ setminus (Inj0 x0) (Sing 0)}.
Apply ReplI with
setminus (Inj0 x0) (Sing 0),
Unj,
Inj1 x1.
Apply setminusI with
Inj0 x0,
Sing 0,
Inj1 x1 leaving 2 subgoals.
Apply ReplI with
x0,
Inj1,
x1.
The subproof is completed by applying H0.
The subproof is completed by applying Inj1NE2 with x1.