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.

Let x14 of type ι be given.

Assume H15: x14 ∈ x0.

Apply setminusE with x1, Sing x3, x4, 25ef3.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 ⟶ ∀ x15 : ο . (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x2 x11 x3 ⟶ not (x2 x12 x3) ⟶ not (x2 x13 x3) ⟶ not (x2 x14 x3) ⟶ x15) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x2 x11 x3 ⟶ not (x2 x12 x3) ⟶ not (x2 x13 x3) ⟶ not (x2 x14 x3) ⟶ x15) ⟶ x15 leaving 2 subgoals.

Apply H4 with x4.

The subproof is completed by applying H5.

Assume H16: x4 ∈ x1.

Assume H17: nIn x4 (Sing x3).

Apply setminusE with x1, Sing x3, x5, ... ⟶ ∀ x15 : ο . (... ⟶ ... ⟶ ... ⟶ ... ⟶ not ... ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x2 x11 x3 ⟶ not (x2 x12 x3) ⟶ not (x2 x13 x3) ⟶ not (x2 x14 x3) ⟶ x15) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x2 x11 x3 ⟶ not (x2 x12 x3) ⟶ not (x2 x13 x3) ⟶ not (x2 x14 x3) ⟶ x15) ⟶ x15 leaving 2 subgoals.

...

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Proofgold Proof