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.
Assume H2: x2 ∈ x0.
Let x3 of type ι be given.
Let x4 of type ι be given.
Assume H5: x3 = x4 ⟶ ∀ x5 : ο . x5.
Assume H6: x1 x3 x4.
Apply SepE with
x0,
λ x5 . and (x2 = x5 ⟶ ∀ x6 : ο . x6) (x1 x2 x5),
x3,
False leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H7: x3 ∈ x0.
Assume H8:
and (x2 = x3 ⟶ ∀ x5 : ο . x5) (x1 x2 x3).
Apply H8 with
False.
Assume H9: x2 = x3 ⟶ ∀ x5 : ο . x5.
Assume H10: x1 x2 x3.
Apply SepE with
x0,
λ x5 . and (x2 = x5 ⟶ ∀ x6 : ο . x6) (x1 x2 x5),
x4,
False leaving 2 subgoals.
The subproof is completed by applying H4.
Assume H11: x4 ∈ x0.
Assume H12:
and (x2 = x4 ⟶ ∀ x5 : ο . x5) (x1 x2 x4).
Apply H12 with
False.
Assume H13: x2 = x4 ⟶ ∀ x5 : ο . x5.
Assume H14: x1 x2 x4.
Apply H1 with
SetAdjoin (UPair x2 x3) x4 leaving 3 subgoals.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with
x2,
x3,
x4,
λ x5 . x5 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
The subproof is completed by applying H11.
Apply unknownprop_8a21f6cb5fc1714044127ec01eb34af4a43c7190a9ab55c5830d9c24f7e274f6 with
SetAdjoin (UPair x2 x3) x4,
x2,
x3,
x4 leaving 6 subgoals.
The subproof is completed by applying unknownprop_2f981bb386e15ae80933d34ec7d4feaabeedc598a3b07fb73b422d0a88302c67 with x2, x3, x4.
The subproof is completed by applying unknownprop_91640ab91f642c55f5e5a7feb12af7896a6f3419531543b011f7b54a888153d1 with x2, x3, x4.
The subproof is completed by applying unknownprop_ca66642b4e7ed479322d8970220318ddbb0c129adc66c35d9ce66f8223608389 with x2, x3, x4.
The subproof is completed by applying H9.
The subproof is completed by applying H13.
The subproof is completed by applying H5.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with
x2,
x3,
x4,
λ x5 . ∀ x6 . x6 ∈ SetAdjoin (UPair x2 x3) x4 ⟶ (x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ x1 x5 x6 leaving 3 subgoals.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with
x2,
x3,
x4,
λ x5 . (x2 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1 x2 x5 leaving 3 subgoals.
Assume H15: x2 = x2 ⟶ ∀ x5 : ο . x5.
Apply FalseE with
x1 x2 x2.
Apply H15.
Let x5 of type ι → ι → ο be given.
Assume H16: x5 x2 x2.
The subproof is completed by applying H16.
Assume H15: x2 = x3 ⟶ ∀ x5 : ο . x5.
The subproof is completed by applying H10.
Assume H15: x2 = x4 ⟶ ∀ x5 : ο . x5.
The subproof is completed by applying H14.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with
x2,
x3,
x4,
λ x5 . (x3 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1 x3 x5 leaving 3 subgoals.
Assume H15: x3 = x2 ⟶ ∀ x5 : ο . x5.
Apply H0 with
x2,
x3.
The subproof is completed by applying H10.
Assume H15: x3 = x3 ⟶ ∀ x5 : ο . x5.
Apply FalseE with
x1 x3 x3.
Apply H15.
Let x5 of type ι → ι → ο be given.
Assume H16: x5 x3 x3.
The subproof is completed by applying H16.
Assume H15: x3 = x4 ⟶ ∀ x5 : ο . x5.
The subproof is completed by applying H6.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with
x2,
x3,
x4,
λ x5 . (x4 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1 x4 x5 leaving 3 subgoals.