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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 ∈ {x3 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x3) (DirGraphOutNeighbors u18 x0 x1)) u1}.
Let x3 of type ι be given.
Assume H5: x3 ∈ {x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x4) (DirGraphOutNeighbors u18 x0 x1)) u1}.
Assume H6: 4b3fa.. x0 x1 x2 = 4b3fa.. x0 x1 x3.
Apply binintersectE with DirGraphOutNeighbors u18 x0 x2, DirGraphOutNeighbors u18 x0 x1, 4b3fa.. x0 x1 x2, x2 = x3 leaving 2 subgoals.
Apply unknownprop_189d30706526e416bb923c2b4c5eecdaf2e5f4a9bfc47487e0d636eccfc5f8f7 with x0, x1, x2 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 H4.
Assume H7: 4b3fa.. x0 x1 x2 ∈ DirGraphOutNeighbors u18 x0 x2.
Assume H8: 4b3fa.. x0 x1 x2 ∈ DirGraphOutNeighbors u18 x0 x1.
Apply unknownprop_14e8b1cab6218ae78b591f7980f02116989e69b3919a40df92b352b5ca17bd33 with x0, x1, 4b3fa.. x0 x1 x2, x2, x3 leaving 9 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 H8.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply SepE2 with u18, λ x4 . and (x2 = x4 ⟶ ∀ x5 : ο . x5) (x0 x2 x4), 4b3fa.. x0 x1 x2, x0 x2 (4b3fa.. x0 x1 x2) leaving 2 subgoals.
The subproof is completed by applying H7.
Assume H9: x2 = 4b3fa.. x0 x1 x2 ⟶ ∀ x4 : ο . x4.
Assume H10: x0 x2 (4b3fa.. x0 x1 x2).
The subproof is completed by applying H10.
Apply H6 with λ x4 x5 . x0 x3 x5.
Apply SepE2 with u18, λ x4 . and (x3 = x4 ⟶ ∀ x5 : ο . x5) (x0 x3 x4), 4b3fa.. x0 x1 x3, x0 x3 (4b3fa.. x0 x1 x3) leaving 2 subgoals.
Apply binintersectE1 with DirGraphOutNeighbors u18 x0 x3, DirGraphOutNeighbors u18 x0 x1, 4b3fa.. x0 x1 x3.
Apply unknownprop_189d30706526e416bb923c2b4c5eecdaf2e5f4a9bfc47487e0d636eccfc5f8f7 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 H5.
Assume H9: x3 = 4b3fa.. x0 x1 x3 ⟶ ∀ x4 : ο . x4.
Assume H10: x0 x3 (4b3fa.. x0 x1 x3).
The subproof is completed by applying H10.
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