Let x0 of type ι → ι → ο be given.
Assume H0: ∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1.
Let x1 of type ι be given.
Assume H1: x1 ∈ 6.
Let x2 of type ι be given.
Assume H2: x2 ∈ 6.
Let x3 of type ι be given.
Assume H3: x3 ∈ 6.
Assume H4: x1 = x2 ⟶ ∀ x4 : ο . x4.
Assume H5: x1 = x3 ⟶ ∀ x4 : ο . x4.
Assume H6: x2 = x3 ⟶ ∀ x4 : ο . x4.
Assume H7: x0 x1 x2.
Assume H8: x0 x1 x3.
Assume H9: x0 x2 x3.
Let x4 of type ο be given.
Assume H10:
∀ x5 . and (x5 ⊆ 6) (and (equip 3 x5) (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 . x7 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x8 : ο . x8) ⟶ x0 x6 x7)) ⟶ x4.
Apply H10 with
SetAdjoin (UPair x1 x2) x3.
Apply andI with
SetAdjoin (UPair x1 x2) x3 ⊆ 6,
and (equip 3 (SetAdjoin (UPair x1 x2) x3)) (∀ x5 . x5 ∈ SetAdjoin (UPair x1 x2) x3 ⟶ ∀ x6 . x6 ∈ SetAdjoin (UPair x1 x2) x3 ⟶ (x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ x0 x5 x6) leaving 2 subgoals.
Let x5 of type ι be given.
Apply binunionE with
UPair x1 x2,
Sing x3,
x5,
x5 ∈ 6 leaving 3 subgoals.
The subproof is completed by applying H11.
Assume H12:
x5 ∈ UPair x1 x2.
Apply UPairE with
x5,
x1,
x2,
x5 ∈ 6 leaving 3 subgoals.
The subproof is completed by applying H12.
Assume H13: x5 = x1.
Apply H13 with
λ x6 x7 . x7 ∈ 6.
The subproof is completed by applying H1.
Assume H13: x5 = x2.
Apply H13 with
λ x6 x7 . x7 ∈ 6.
The subproof is completed by applying H2.
Assume H12:
x5 ∈ Sing x3.
Apply SingE with
x3,
x5,
λ x6 x7 . x7 ∈ 6 leaving 2 subgoals.
The subproof is completed by applying H12.
The subproof is completed by applying H3.
Apply andI with
equip 3 (SetAdjoin (UPair x1 x2) x3),
∀ x5 . x5 ∈ SetAdjoin (UPair x1 x2) x3 ⟶ ∀ x6 . x6 ∈ SetAdjoin (UPair x1 x2) x3 ⟶ (x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ x0 x5 x6 leaving 2 subgoals.
Apply unknownprop_637144c754e35176e5f73e9789b35a2d801de40f26463f5ae01a3b9c5aad6047 with
x1,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Let x5 of type ι be given.
Let x6 of type ι be given.
Apply binunionE with
UPair x1 x2,
Sing x3,
x5,
(x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ x0 x5 x6 leaving 3 subgoals.
The subproof is completed by applying H11.
Assume H13:
x5 ∈ UPair x1 x2.
Apply UPairE with
x5,
x1,
x2,
(x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ x0 x5 x6 leaving 3 subgoals.
The subproof is completed by applying H13.
Assume H14: x5 = x1.
Apply H14 with
λ x7 x8 . (x8 = x6 ⟶ ∀ x9 : ο . x9) ⟶ x0 x8 x6.
Apply binunionE with
UPair ... ...,
...,
...,
... leaving 3 subgoals.