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.

Apply setminusE with x1, Sing x3, x4, 2bad0.. x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 ⟶ ∀ x13 : ο . (∀ x14 . x14 ∈ x0 ⟶ ∀ x15 . x15 ∈ x0 ⟶ ∀ x16 . x16 ∈ x0 ⟶ ∀ x17 . x17 ∈ x0 ⟶ ∀ x18 . x18 ∈ x0 ⟶ ∀ x19 . x19 ∈ x0 ⟶ ∀ x20 . x20 ∈ x0 ⟶ ∀ x21 . x21 ∈ x0 ⟶ ∀ x22 . x22 ∈ x0 ⟶ d0208.. x2 x14 x15 x16 x3 x17 x18 x19 x20 x21 x22 ⟶ x13) ⟶ (∀ x14 . x14 ∈ x0 ⟶ ∀ x15 . x15 ∈ x0 ⟶ ∀ x16 . x16 ∈ x0 ⟶ ∀ x17 . x17 ∈ x0 ⟶ ∀ x18 . x18 ∈ x0 ⟶ ∀ x19 . x19 ∈ x0 ⟶ ∀ x20 . x20 ∈ x0 ⟶ ∀ x21 . x21 ∈ x0 ⟶ ∀ x22 . x22 ∈ x0 ⟶ ea11f.. x2 x14 x15 x16 x17 x18 x19 x20 x21 x3 x22 ⟶ x13) ⟶ x13 leaving 2 subgoals.

Apply H4 with x4.

The subproof is completed by applying H5.

Assume H14: x4 ∈ x1.

Assume H15: nIn x4 (Sing x3).

Apply setminusE with x1, Sing x3, x5, ... ⟶ ∀ x13 : ο . (∀ x14 . ... ⟶ ∀ x15 . ... ⟶ ∀ x16 . ... ⟶ ∀ x17 . ... ⟶ ∀ x18 . ... ⟶ ∀ x19 . ... ⟶ ∀ x20 . x20 ∈ ... ⟶ ∀ x21 . x21 ∈ x0 ⟶ ∀ x22 . x22 ∈ x0 ⟶ d0208.. x2 x14 x15 x16 x3 x17 x18 x19 x20 x21 x22 ⟶ x13) ⟶ (∀ x14 . x14 ∈ x0 ⟶ ∀ x15 . x15 ∈ x0 ⟶ ∀ x16 . x16 ∈ x0 ⟶ ∀ x17 . x17 ∈ x0 ⟶ ∀ x18 . x18 ∈ x0 ⟶ ∀ x19 . x19 ∈ x0 ⟶ ∀ x20 . x20 ∈ x0 ⟶ ∀ x21 . x21 ∈ x0 ⟶ ∀ x22 . x22 ∈ x0 ⟶ ea11f.. x2 x14 x15 x16 x17 x18 x19 x20 x21 x3 x22 ⟶ x13) ⟶ x13 leaving 2 subgoals.

...

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Proofgold Proof