Let x0 of type ι → ι → ο be given.

Assume H0: ∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1.

Assume H1: ∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3).

Assume H2: ∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3)).

Let x1 of type ι be given.

Assume H3: x1 ∈ u18.

Let x2 of type ι be given.

Assume H4: x2 ∈ DirGraphOutNeighbors u18 x0 x1.

Let x3 of type ι be given.

Let x4 of type ι be given.

Assume H7: x0 x3 x2.

Assume H8: x0 x4 x2.

Apply dneg with x3 = x4.

Assume H9: x3 = x4 ⟶ ∀ x5 : ο . x5.

The subproof is completed by applying H5.

Apply setminusE with u18, binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1), x3, False leaving 2 subgoals.

The subproof is completed by applying H10.

Assume H12: x3 ∈ u18.

The subproof is completed by applying H6.

Apply setminusE with u18, binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1), x4, False leaving 2 subgoals.

The subproof is completed by applying H14.

Assume H16: x4 ∈ u18.

Assume H17: nIn x4 (binunion ... ...).

...

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Proofgold Proof