Let x0 of type ι → ι → ο be given.
Assume H0: ∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1.
Assume H2:
not (or (∃ x1 . and (x1 ⊆ u9) (and (equip u3 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3))) (∃ x1 . and (x1 ⊆ u9) (and (equip u4 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3))))).
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H6: x1 = x2 ⟶ ∀ x4 : ο . x4.
Assume H7: x1 = x3 ⟶ ∀ x4 : ο . x4.
Assume H8: x2 = x3 ⟶ ∀ x4 : ο . x4.
Assume H9: x0 x1 x2.
Assume H10: x0 x1 x3.
Let x4 of type ο be given.
Assume H11:
∀ x5 . x5 ∈ u9 ⟶ (x1 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x0 x1 x5 ⟶ not (x0 x2 x5) ⟶ not (x0 x3 x5) ⟶ (∀ x6 . x6 ∈ u9 ⟶ x0 x1 x6 ⟶ x6 ∈ SetAdjoin (SetAdjoin (UPair x1 x2) x3) x5) ⟶ x4.
Apply unknownprop_5df8c11eede12b7d829e2a0b95c8033f28b954d28a1a477c3fa227324f16bb6c with
x0,
x1,
x2,
x4 leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H6.
The subproof is completed by applying H9.
Let x5 of type ι be given.
Let x6 of type ι be given.
Assume H14: x1 = x5 ⟶ ∀ x7 : ο . x7.
Assume H15: x1 = x6 ⟶ ∀ x7 : ο . x7.
Assume H16: x2 = x5 ⟶ ∀ x7 : ο . x7.
Assume H17: x2 = x6 ⟶ ∀ x7 : ο . x7.
Assume H18: x5 = x6 ⟶ ∀ x7 : ο . x7.
Assume H19: x0 x1 x5.
Assume H20: x0 x1 x6.
Assume H21:
not (x0 x2 x5).
Assume H22:
not (x0 x2 x6).
Assume H23:
not (x0 x5 x6).
Apply L25 leaving 2 subgoals.
Assume H26: x3 = x5.
Apply H26 with
λ x7 x8 . ∀ x9 . x9 ∈ u9 ⟶ x0 x1 x9 ⟶ x9 ∈ SetAdjoin (SetAdjoin (UPair x1 x2) x8) x6.
The subproof is completed by applying H24.
Apply H11 with
x6 leaving 8 subgoals.
The subproof is completed by applying H13.
The subproof is completed by applying H15.
The subproof is completed by applying H17.
The subproof is completed by applying L27.
The subproof is completed by applying H20.
The subproof is completed by applying H22.
The subproof is completed by applying L28.
The subproof is completed by applying L29.