Let x0 of type ι be given.

Let x1 of type ι → ι → ο be given.

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

Assume H1: ∀ x2 . x2 ⊆ x0 ⟶ atleastp u3 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1 x3 x4).

Assume H2: ∀ x2 . x2 ⊆ x0 ⟶ atleastp u6 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4)).

Let x2 of type ι be given.

Let x3 of type ι be given.

Assume H3: x2 ∈ x0.

Let x4 of type ι be given.

Assume H4: nat_p x4.

Assume H5: ∀ x5 . x5 ∈ x0 ⟶ (x5 = x2 ⟶ ∀ x6 : ο . x6) ⟶ not (x1 x5 x2) ⟶ or (equip (binintersect (DirGraphOutNeighbors x0 x1 x5) (DirGraphOutNeighbors x0 x1 x2)) x4) (equip (binintersect (DirGraphOutNeighbors x0 x1 x5) (DirGraphOutNeighbors x0 x1 x2)) (ordsucc x4)).

Assume H6: ∀ x5 . x5 ∈ {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x2) (Sing x2))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x2)) x4} ⟶ not (x1 x3 x5).

Assume H7: ∀ x5 . x5 ∈ setminus (DirGraphOutNeighbors x0 x1 x3) (Sing x2) ⟶ not (x1 x2 x5).

Apply unknownprop_5f1f39832970a5af1d79c81e14d5505638e1408e0e92f90fda2e62e7a0bce390 with x0, x1, x2, x3, x4 leaving 6 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

The subproof is completed by applying H5.

The subproof is completed by applying H6.

The subproof is completed by applying H7.

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Proofgold Proof