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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.
Let x3 of type ι be given.
Assume H5: 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, f14ce.. x0 x1 x3 ∈ setminus (DirGraphOutNeighbors u18 x0 x1) (Sing x2) leaving 2 subgoals.
The subproof is completed by applying H5.
Assume H6: x3 ∈ {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u2}.
Assume H7: 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, f14ce.. x0 x1 x3 ∈ setminus (DirGraphOutNeighbors u18 x0 x1) (Sing x2) leaving 2 subgoals.
The subproof is completed by applying H6.
Assume H8: x3 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1)).
Assume H9: equip (binintersect (DirGraphOutNeighbors u18 x0 x3) (DirGraphOutNeighbors u18 x0 x1)) u2.
Apply binintersectE with DirGraphOutNeighbors u18 x0 x3, DirGraphOutNeighbors u18 x0 x1, f14ce.. x0 x1 x3, f14ce.. x0 x1 x3 ∈ setminus (DirGraphOutNeighbors u18 x0 x1) (Sing x2) leaving 2 subgoals.
Apply unknownprop_9e26c72c445a0733558f93c30d08b60533b6215eef57de5a32349b2abb17cf19 with x0, x1, x3 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H6.
Assume H10: f14ce.. x0 x1 x3 ∈ DirGraphOutNeighbors u18 x0 x3.
Assume H11: f14ce.. x0 ... ... ∈ ....
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