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.
Assume H3: x2 ⊆ x0.
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.
Assume H5: x4 = x3 ⟶ ∀ x7 : ο . x7.
Assume H6: x5 = x3 ⟶ ∀ x7 : ο . x7.
Assume H7: x6 = x3 ⟶ ∀ x7 : ο . x7.
Assume H8: x5 = x4 ⟶ ∀ x7 : ο . x7.
Assume H9: x6 = x4 ⟶ ∀ x7 : ο . x7.
Assume H10: x6 = x5 ⟶ ∀ x7 : ο . x7.
Assume H11: x1 x3 x4.
Assume H12: x1 x4 x5.
Assume H13: x1 x5 x6.
Assume H14: x1 x6 x3.
Apply unknownprop_69a9db44595939e1bddcd5bc279c628551a6750716ca8682b12653ef28485775 with
x0,
x1,
x2,
x3,
x4,
x5,
x6 leaving 16 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.