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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 ∈ DirGraphOutNeighbors u18 x0 x1.
Assume H6: x2 = x3 ⟶ ∀ x4 : ο . x4.
Let x4 of type ι be given.
Assume H7: x4 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1)).
Let x5 of type ι be given.
Assume H8: x5 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1)).
Assume H9: x0 x4 x2.
Assume H10: x0 x5 x2.
Assume H11: x0 x4 x3.
Assume H12: x0 x5 x3.
Apply dneg with x4 = x5.
Assume H13: x4 = x5 ⟶ ∀ x6 : ο . x6.
Apply setminusE with u18, binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1), x4, False leaving 2 subgoals.
The subproof is completed by applying H7.
Assume H14: x4 ∈ u18.
Assume H15: nIn x4 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1)).
Apply setminusE with u18, binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1), x5, False leaving 2 subgoals.
The subproof is completed by applying H8.
Assume H16: x5 ∈ u18.
Assume H17: nIn x5 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1)).
Apply SepE with u18, λ x6 . and (x1 = x6 ⟶ ∀ x7 : ο . x7) (x0 x1 x6), x2, False leaving 2 subgoals.
The subproof is completed by applying H4.
Assume H18: x2 ∈ u18.
Assume H19: and (x1 = x2 ⟶ ∀ x6 : ο . x6) (x0 x1 x2).
Apply H19 with False.
Assume H20: x1 = x2 ⟶ ∀ x6 : ο . x6.
Assume H21: x0 x1 x2.
Apply SepE with u18, λ x6 . and (x1 = x6 ⟶ ∀ x7 : ο . x7) (x0 x1 x6), x3, False leaving 2 subgoals.
The subproof is completed by applying H5.
Assume H22: x3 ∈ u18.
Assume H23: and (x1 = x3 ⟶ ∀ x6 : ο . x6) (x0 x1 x3).
Apply H23 with False.
Assume H24: x1 = x3 ⟶ ∀ x6 : ο . x6.
Assume H25: x0 x1 x3.
Apply unknownprop_8a6bdce060c93f04626730b6e01b099cc0487102a697e253c81b39b9a082262d with u2 leaving 2 subgoals.
The subproof is completed by applying nat_2.
Apply atleastp_tra with u3, SetAdjoin (UPair x1 x4) x5, u2 leaving 2 subgoals.
Apply unknownprop_8a21f6cb5fc1714044127ec01eb34af4a43c7190a9ab55c5830d9c24f7e274f6 with SetAdjoin (UPair x1 x4) x5, x1, x4, x5 leaving 6 subgoals.
The subproof is completed by applying unknownprop_2f981bb386e15ae80933d34ec7d4feaabeedc598a3b07fb73b422d0a88302c67 with x1, x4, x5.
The subproof is completed by applying unknownprop_91640ab91f642c55f5e5a7feb12af7896a6f3419531543b011f7b54a888153d1 with x1, x4, x5.
The subproof is completed by applying unknownprop_ca66642b4e7ed479322d8970220318ddbb0c129adc66c35d9ce66f8223608389 with x1, x4, x5.
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