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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.
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).
Assume H7: 31e20.. x0 x1 (4b3fa.. x0 x1 x3) = 31e20.. x0 x1 (f14ce.. x0 x1 x3).
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, False leaving 2 subgoals.
The subproof is completed by applying H6.
Assume H8: x3 ∈ {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u2}.
Assume H9: nIn x3 (DirGraphOutNeighbors u18 x0 x2).
Apply unknownprop_319dc69e13a657ae6992f9456b32382c784ca132f3312b0bba7ee0e66ca50b9d with x0, x1, x2, 4b3fa.. x0 x1 x3, False leaving 8 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 H4.
The subproof is completed by applying H5.
Apply unknownprop_00d76dfe53c0efe0a1f290756a6cc73e1988f17bfbfcefb07db11203a1549140 with x0, x1, x2, x3 leaving 6 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 H4.
Apply setminusI with setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1)), DirGraphOutNeighbors u18 x0 x2, x3 leaving 2 subgoals.
Apply SepE1 with setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1)), λ x4 . equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u2, x3.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Assume H10: and (31e20.. x0 x1 (4b3fa.. x0 x1 x3) ∈ {x4 ∈ setminus u18 (binunion ... ...)|...}) ....
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