Apply finite_ind with λ x0 . ∀ x1 . x1 ⊆ x0 ⟶ finite x1 leaving 2 subgoals.

Let x0 of type ι be given.

Assume H0: x0 ⊆ 0.

Apply Empty_Subq_eq with x0, λ x1 x2 . finite x2 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying finite_Empty.

Let x0 of type ι be given.

Let x1 of type ι be given.

Assume H0: finite x0.

Assume H1: nIn x1 x0.

Assume H2: ∀ x2 . x2 ⊆ x0 ⟶ finite x2.

Let x2 of type ι be given.

Assume H3: x2 ⊆ binunion x0 (Sing x1).

Apply xm with x1 ∈ x2, finite x2 leaving 2 subgoals.

Assume H4: x1 ∈ x2.

Apply set_ext with x2, binunion (setminus x2 (Sing x1)) (Sing x1), λ x3 x4 . finite x4 leaving 3 subgoals.

Let x3 of type ι be given.

Assume H5: x3 ∈ x2.

Apply xm with x3 ∈ Sing x1, x3 ∈ binunion (setminus x2 (Sing x1)) (Sing x1) leaving 2 subgoals.

The subproof is completed by applying binunionI2 with setminus x2 (Sing x1), Sing x1, x3.

Assume H6: nIn x3 (Sing x1).

Apply binunionI1 with setminus x2 (Sing x1), Sing x1, x3.

Apply setminusI with x2, Sing x1, x3 leaving 2 subgoals.

The subproof is completed by applying H5.

The subproof is completed by applying H6.

Let x3 of type ι be given.

Assume H5: x3 ∈ binunion (setminus x2 (Sing x1)) (Sing x1).

Apply binunionE with setminus x2 (Sing x1), Sing x1, x3, x3 ∈ x2 leaving 3 subgoals.

The subproof is completed by applying H5.

The subproof is completed by applying setminusE1 with x2, Sing x1, x3.

Assume H6: x3 ∈ Sing x1.

Apply SingE with x1, x3, λ x4 x5 . x5 ∈ x2 leaving 2 subgoals.

The subproof is completed by applying H6.

The subproof is completed by applying H4.

Apply adjoin_finite with setminus x2 (Sing x1), x1.

Apply H2 with setminus x2 (Sing x1).

Let x3 of type ι be given.

Assume H5: x3 ∈ setminus x2 (Sing x1).

Apply setminusE with x2, Sing x1, x3, x3 ∈ x0 leaving 2 subgoals.

The subproof is completed by applying H5.

Assume H6: x3 ∈ x2.

Assume H7: nIn x3 (Sing x1).

Apply binunionE with x0, Sing x1, x3, x3 ∈ x0 leaving 3 subgoals.

Apply H3 with x3.

The subproof is completed by applying H6.

Assume H8: x3 ∈ x0.

The subproof is completed by applying H8.

Assume H8: x3 ∈ Sing x1.

Apply FalseE with x3 ∈ x0.

Apply H7.

The subproof is completed by applying H8.

Assume H4: nIn x1 x2.

Apply H2 with x2.

Let x3 of type ι be given.

Assume H5: x3 ∈ x2.

Apply binunionE with x0, Sing x1, x3, x3 ∈ x0 leaving 3 subgoals.

Apply H3 with x3.

The subproof is completed by applying H5.

Assume H6: x3 ∈ x0.

The subproof is completed by applying H6.

Assume H6: x3 ∈ Sing x1.

Apply FalseE with x3 ∈ x0.

Apply H4.

Apply SingE with x1, x3, λ x4 x5 . x4 ∈ x2 leaving 2 subgoals.

The subproof is completed by applying H6.

The subproof is completed by applying H5.

■

Proofgold Proof