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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Assume H0: ∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2.
Assume H1: ∀ x2 . x2 ⊆ x0 ⟶ atleastp u3 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1 x3 x4).
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H2: x2 ∈ x0.
Assume H3: x3 ∈ DirGraphOutNeighbors x0 x1 x2.
Assume H4: x2 = x3 ⟶ ∀ x4 : ο . x4.
Assume H5: x1 x2 x3.
Let x4 of type ι be given.
Assume H6: x4 ∈ setminus (DirGraphOutNeighbors x0 x1 x3) (Sing x2).
Assume H7: x1 x2 x4.
Apply setminusE with DirGraphOutNeighbors x0 x1 x3, Sing x2, x4, False leaving 2 subgoals.
The subproof is completed by applying H6.
Assume H8: x4 ∈ DirGraphOutNeighbors x0 x1 x3.
Assume H9: nIn x4 (Sing x2).
Apply SepE with x0, λ x5 . and (x3 = x5 ⟶ ∀ x6 : ο . x6) (x1 x3 x5), x4, False leaving 2 subgoals.
The subproof is completed by applying H8.
Assume H10: x4 ∈ x0.
Assume H11: and (x3 = x4 ⟶ ∀ x5 : ο . x5) (x1 x3 x4).
Apply H11 with False.
Assume H12: x3 = x4 ⟶ ∀ x5 : ο . x5.
Assume H13: x1 x3 x4.
Apply H1 with SetAdjoin (UPair x3 x4) x2 leaving 3 subgoals.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with x3, x4, x2, λ x5 . x5 ∈ x0 leaving 3 subgoals.
Apply SepE1 with x0, λ x5 . and (x2 = x5 ⟶ ∀ x6 : ο . x6) (x1 x2 x5), x3.
The subproof is completed by applying H3.
The subproof is completed by applying H10.
The subproof is completed by applying H2.
Apply unknownprop_8a21f6cb5fc1714044127ec01eb34af4a43c7190a9ab55c5830d9c24f7e274f6 with SetAdjoin (UPair x3 x4) x2, x3, x4, x2 leaving 6 subgoals.
The subproof is completed by applying unknownprop_2f981bb386e15ae80933d34ec7d4feaabeedc598a3b07fb73b422d0a88302c67 with x3, x4, x2.
The subproof is completed by applying unknownprop_91640ab91f642c55f5e5a7feb12af7896a6f3419531543b011f7b54a888153d1 with x3, x4, x2.
The subproof is completed by applying unknownprop_ca66642b4e7ed479322d8970220318ddbb0c129adc66c35d9ce66f8223608389 with x3, x4, x2.
The subproof is completed by applying H12.
Apply neq_i_sym with x2, x3.
The subproof is completed by applying H4.
Assume H14: x4 = x2.
Apply H9.
Apply H14 with λ x5 x6 . x6 ∈ Sing x2.
The subproof is completed by applying SingI with x2.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with x3, x4, x2, λ x5 . ∀ x6 . x6 ∈ SetAdjoin (UPair x3 x4) x2 ⟶ (x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ x1 x5 x6 leaving 3 subgoals.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with x3, x4, x2, λ x5 . (x3 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1 x3 x5 leaving 3 subgoals.
Assume H14: x3 = x3 ⟶ ∀ x5 : ο . x5.
Apply FalseE with x1 x3 x3.
Apply H14.
Let x5 of type ι → ι → ο be given.
Assume H15: x5 x3 x3.
The subproof is completed by applying H15.
Assume H14: x3 = x4 ⟶ ∀ x5 : ο . x5.
The subproof is completed by applying H13.
Assume H14: x3 = x2 ⟶ ∀ x5 : ο . x5.
Apply H0 with x2, x3.
The subproof is completed by applying H5.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with x3, x4, x2, λ x5 . (x4 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1 x4 x5 leaving 3 subgoals.
Assume H14: x4 = x3 ⟶ ∀ x5 : ο . x5.
Apply H0 with x3, x4.
The subproof is completed by applying H13.
Assume H14: x4 = x4 ⟶ ∀ x5 : ο . x5.
Apply FalseE with x1 x4 x4.
Apply H14.
Let x5 of type ι → ι → ο be given.
Assume H15: x5 x4 x4.
The subproof is completed by applying H15.
Assume H14: ... = ... ⟶ ∀ x5 : ο . x5.
...
...
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