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.

Claim L4: ...

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Claim L5: ...

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Claim L6: ...

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Apply L5 with and (equip {x2 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x2) (DirGraphOutNeighbors u18 x0 x1)) u1} u4) (equip {x2 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x2) (DirGraphOutNeighbors u18 x0 x1)) u2} u8).

Let x2 of type ι be given.

Assume H7: (λ x3 . and (x3 ∈ omega) (equip {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u1} x3)) x2.

Apply H7 with and (equip {x3 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x3) (DirGraphOutNeighbors u18 x0 x1)) u1} u4) (equip {x3 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x3) (DirGraphOutNeighbors u18 x0 x1)) u2} u8).

Assume H8: x2 ∈ omega.

Assume H9: equip {x3 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x3) (DirGraphOutNeighbors u18 x0 x1)) u1} x2.

Apply L6 with and (equip {x3 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x3) (DirGraphOutNeighbors u18 x0 x1)) u1} u4) (equip {x3 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x3) (DirGraphOutNeighbors u18 x0 x1)) u2} ...).

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