Apply In_ind with λ x0 . Unj (Inj1 x0) = x0.

Let x0 of type ι be given.

Assume H0: ∀ x1 . x1 ∈ x0 ⟶ Unj (Inj1 x1) = x1.

Apply Unj_eq with Inj1 x0, λ x1 x2 . x2 = x0.

Apply set_ext with {Unj x1|x1 ∈ setminus (Inj1 x0) (Sing 0)}, x0 leaving 2 subgoals.

Let x1 of type ι be given.

Assume H1: x1 ∈ {Unj x2|x2 ∈ setminus (Inj1 x0) (Sing 0)}.

Apply ReplE_impred with setminus (Inj1 x0) (Sing 0), Unj, x1, x1 ∈ x0 leaving 2 subgoals.

The subproof is completed by applying H1.

Let x2 of type ι be given.

Assume H2: x2 ∈ setminus (Inj1 x0) (Sing 0).

Assume H3: x1 = Unj x2.

Apply H3 with λ x3 x4 . x4 ∈ x0.

Apply setminusE with Inj1 x0, Sing 0, x2, Unj x2 ∈ x0 leaving 2 subgoals.

The subproof is completed by applying H2.

Assume H4: x2 ∈ Inj1 x0.

Assume H5: nIn x2 (Sing 0).

Apply Inj1E with x0, x2, Unj x2 ∈ x0 leaving 3 subgoals.

The subproof is completed by applying H4.

Assume H6: x2 = 0.

Apply FalseE with Unj x2 ∈ x0.

Apply H5.

Apply H6 with λ x3 x4 . x4 ∈ Sing 0.

The subproof is completed by applying SingI with 0.

Assume H6: ∃ x3 . and (x3 ∈ x0) (x2 = Inj1 x3).

Apply exandE_i with λ x3 . x3 ∈ x0, λ x3 . x2 = Inj1 x3, Unj x2 ∈ x0 leaving 2 subgoals.

The subproof is completed by applying H6.

Let x3 of type ι be given.

Assume H7: x3 ∈ x0.

Assume H8: x2 = Inj1 x3.

Apply H8 with λ x4 x5 . Unj x5 ∈ x0.

Apply H0 with x3, λ x4 x5 . x5 ∈ x0 leaving 2 subgoals.

The subproof is completed by applying H7.

Let x1 of type ι be given.

Assume H1: x1 ∈ x0.

Apply H0 with x1, λ x2 x3 . x2 ∈ {Unj x4|x4 ∈ setminus (Inj1 x0) (Sing 0)} leaving 2 subgoals.

The subproof is completed by applying H1.

Apply ReplI with setminus (Inj1 x0) (Sing 0), Unj, Inj1 x1.

Apply setminusI with Inj1 x0, Sing 0, Inj1 x1 leaving 2 subgoals.

Apply Inj1I2 with x0, x1.

The subproof is completed by applying H1.

The subproof is completed by applying Inj1NE2 with x1.

■

Proofgold Proof