Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι → ι → ο be given.

Assume H0: ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3.

Assume H1: 4402e.. x1 x2.

Assume H2: cf2df.. x1 x2.

Let x3 of type ι be given.

Assume H3: x3 ∈ x1.

Assume H4: x0 ⊆ setminus x1 (Sing x3).

Let x4 of type ι be given.

Assume H5: x4 ∈ x0.

Let x5 of type ι be given.

Assume H6: x5 ∈ x0.

Let x6 of type ι be given.

Assume H7: x6 ∈ x0.

Let x7 of type ι be given.

Assume H8: x7 ∈ x0.

Let x8 of type ι be given.

Assume H9: x8 ∈ x0.

Let x9 of type ι be given.

Assume H10: x9 ∈ x0.

Let x10 of type ι be given.

Assume H11: x10 ∈ x0.

Let x11 of type ι be given.

Assume H12: x11 ∈ x0.

Let x12 of type ι be given.

Assume H13: x12 ∈ x0.

Let x13 of type ι be given.

Assume H14: x13 ∈ x0.

Apply setminusE with x1, Sing x3, x4, f78c3.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ ∀ x14 : ο . x14 leaving 2 subgoals.

Apply H4 with x4.

The subproof is completed by applying H5.

Assume H15: x4 ∈ x1.

Assume H16: nIn x4 (Sing x3).

Apply setminusE with x1, Sing x3, x5, f78c3.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ ∀ x14 : ο . x14 leaving 2 subgoals.

Apply H4 with x5.

The subproof is completed by applying H6.

Assume H17: x5 ∈ x1.

Assume H18: nIn x5 (Sing x3).

Apply setminusE with x1, Sing x3, x6, f78c3.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ ∀ x14 : ο . x14 leaving 2 subgoals.

Apply H4 with x6.

The subproof is completed by applying H7.

Assume H19: x6 ∈ x1.

Assume H20: nIn x6 (Sing x3).

Apply setminusE with x1, Sing x3, x7, f78c3.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ ∀ x14 : ο . x14 leaving 2 subgoals.

Apply H4 with x7.

The subproof is completed by applying H8.

Assume H21: x7 ∈ x1.

Assume H22: nIn x7 (Sing x3).

Apply setminusE with x1, Sing x3, x8, f78c3.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ ∀ x14 : ο . x14 leaving 2 subgoals.

Apply H4 with x8.

The subproof is completed by applying H9.

Assume H23: x8 ∈ x1.

Assume H24: nIn x8 (Sing x3).

Apply setminusE with x1, Sing x3, x9, f78c3.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ ∀ x14 : ο . x14 leaving 2 subgoals.

Apply H4 with x9.

The subproof is completed by applying H10.

Assume H25: x9 ∈ x1.

Assume H26: nIn x9 (Sing x3).

Apply setminusE with x1, Sing x3, x10, f78c3.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ ∀ x14 : ο . x14 leaving 2 subgoals.

Apply H4 with x10.

The subproof is completed by applying H11.

Assume H27: x10 ∈ x1.

Assume H28: nIn x10 (Sing x3).

Apply setminusE with x1, Sing x3, x11, ... ⟶ ∀ x14 : ο . x14 leaving 2 subgoals.

...

■

Proofgold Proof