Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Assume H0: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2.
Let x2 of type ι be given.
Assume H2: x2 ∈ x0.
Let x3 of type ι be given.
Assume H3: x3 ∈ x0.
Let x4 of type ι be given.
Assume H4: x4 ∈ x0.
Assume H5: x2 = x3 ⟶ ∀ x5 : ο . x5.
Assume H6: x2 = x4 ⟶ ∀ x5 : ο . x5.
Assume H7: x3 = x4 ⟶ ∀ x5 : ο . x5.
Assume H8: x1 x2 x3.
Assume H9: x1 x2 x4.
Assume H10: x1 x3 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 H3.
The subproof is completed by applying H4.
Apply equip_atleastp with
u3,
SetAdjoin (UPair x2 x3) x4.
Apply unknownprop_637144c754e35176e5f73e9789b35a2d801de40f26463f5ae01a3b9c5aad6047 with
x2,
x3,
x4 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
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 H11: x2 = x2 ⟶ ∀ x5 : ο . x5.
Apply FalseE with
x1 x2 x2.
Apply H11.
Let x5 of type ι → ι → ο be given.
Assume H12: x5 x2 x2.
The subproof is completed by applying H12.
Assume H11: x2 = x3 ⟶ ∀ x5 : ο . x5.
The subproof is completed by applying H8.
Assume H11: x2 = x4 ⟶ ∀ x5 : ο . x5.
The subproof is completed by applying H9.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with
x2,
x3,
x4,
λ x5 . (x3 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1 x3 x5 leaving 3 subgoals.
Assume H11: x3 = x2 ⟶ ∀ x5 : ο . x5.
Apply H0 with
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H8.
Assume H11: x3 = x3 ⟶ ∀ x5 : ο . x5.
Apply FalseE with
x1 x3 x3.
Apply H11.
Let x5 of type ι → ι → ο be given.
Assume H12: x5 x3 x3.
The subproof is completed by applying H12.
Assume H11: x3 = x4 ⟶ ∀ x5 : ο . x5.
The subproof is completed by applying H10.
Apply unknownprop_434e2e2330a02d70f83efc2b51c595946aeb4462c38cf32d55a1757fe463ba11 with
x2,
x3,
x4,
λ x5 . (x4 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1 x4 x5 leaving 3 subgoals.
Assume H11: x4 = x2 ⟶ ∀ x5 : ο . x5.
Apply H0 with
x2,
x4 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
The subproof is completed by applying H9.
Assume H11: x4 = x3 ⟶ ∀ x5 : ο . x5.
Apply H0 with
x3,
x4 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H10.
Assume H11: x4 = x4 ⟶ ∀ x5 : ο . x5.
Apply FalseE with
x1 x4 x4.
Apply H11.
Let x5 of type ι → ι → ο be given.
Assume H12: x5 x4 x4.
The subproof is completed by applying H12.