Proofgold Proof

pf

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.

Assume H5: ∀ x3 . x3 ∈ {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u1} ⟶ not (x0 x2 x3).

Let x3 of type ι be given.

Assume H6: x3 ∈ setminus {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u2} (DirGraphOutNeighbors u18 x0 x2).

Apply setminusE with {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u2}, DirGraphOutNeighbors u18 x0 x2, x3, 31e20.. x0 x1 (4b3fa.. x0 x1 x3) ∈ {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u1} leaving 2 subgoals.

The subproof is completed by applying H6.

Assume H7: x3 ∈ {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u2}.

Assume H8: nIn x3 (DirGraphOutNeighbors u18 x0 x2).

Apply SepE with setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1)), λ x4 . equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u2, x3, 31e20.. x0 x1 (4b3fa.. x0 x1 x3) ∈ {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u1} leaving 2 subgoals.

The subproof is completed by applying H7.

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