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Proofgold Proof

pf
Let x0 of type ιιο be given.
Assume H0: ∀ x1 x2 . x0 x1 x2x0 x2 x1.
Assume H1: ∀ x1 . x1u18atleastp u3 x1not (∀ x2 . x2x1∀ x3 . x3x1(x2 = x3∀ x4 : ο . x4)x0 x2 x3).
Assume H2: ∀ x1 . x1u18atleastp u6 x1not (∀ x2 . x2x1∀ x3 . x3x1(x2 = x3∀ x4 : ο . x4)not (x0 x2 x3)).
Let x1 of type ι be given.
Assume H3: x1u18.
Apply equip_sym with u6, binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1).
Apply unknownprop_eab190d6552dbda6c7d00c3e93c1ad9385698a8d73462a2a4e5795b67701610d with u5, DirGraphOutNeighbors u18 x0 x1, x1 leaving 2 subgoals.
Assume H4: x1DirGraphOutNeighbors u18 x0 x1.
Apply SepE2 with u18, λ x2 . and (x1 = x2∀ x3 : ο . x3) (x0 x1 x2), x1, False leaving 2 subgoals.
The subproof is completed by applying H4.
Assume H5: x1 = x1∀ x2 : ο . x2.
Apply FalseE with x0 x1 x1False.
Apply H5.
Let x2 of type ιιο be given.
Assume H6: x2 x1 x1.
The subproof is completed by applying H6.
Apply equip_sym with DirGraphOutNeighbors u18 x0 x1, u5.
Apply unknownprop_942eb02a74c10f16602e1ae1f344c3023e05004e91bcaa34f489f7c49867be93 with x0, x1 leaving 4 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.