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).
Assume H2:
∀ x2 . x2 ⊆ x0 ⟶ atleastp u6 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4)).
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Let x9 of type ι be given.
Let x10 of type ι be given.
Let x11 of type ι be given.
Assume H3: x9 ⊆ x0.
Assume H4: x11 ⊆ x0.
Assume H9:
∀ x12 . x12 ∈ x9 ⟶ nIn x12 x8.
Assume H10:
∀ x12 . x12 ∈ x9 ⟶ nIn x12 x11.
Assume H11:
∀ x12 . x12 ∈ x8 ⟶ nIn x12 x11.
Assume H12: x6 ∈ x9.
Assume H13: x7 ∈ x11.
Assume H14: x1 x6 x7.
Let x12 of type ι → ι be given.
Let x13 of type ι → ι be given.
Assume H15: x1 x6 (x12 x6).
Assume H17: ∀ x14 . x14 ∈ x8 ⟶ x12 (x13 x14) = x14.
Apply unknownprop_b452704b69c2690d64de292592f2c296c8b3728aed73fd19657b521cd4c80c64 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13 leaving 17 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
The subproof is completed by applying H14.
The subproof is completed by applying H15.
The subproof is completed by applying H16.
The subproof is completed by applying H17.