Let x0 of type ι → ι → ο be given.
Assume H0: ∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1.
Assume H1:
∀ x1 . x1 ⊆ u17 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3).
Assume H2:
∀ x1 . x1 ⊆ u17 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3)).
Apply H3 with
x0,
False leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H4:
∃ x1 . and (x1 ⊆ u17) (and (atleastp u3 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3)).
Apply H4 with
False.
Let x1 of type ι be given.
Assume H5:
(λ x2 . and (x2 ⊆ u17) (and (atleastp u3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4))) x1.
Apply H5 with
False.
Assume H7:
and (atleastp u3 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3).
Apply H7 with
False.
Assume H9: ∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3.
Apply H1 with
x1 leaving 3 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Assume H4:
∃ x1 . and (x1 ⊆ u17) (and (atleastp u6 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3))).
Apply H4 with
False.
Let x1 of type ι be given.
Assume H5:
(λ x2 . and (x2 ⊆ u17) (and (atleastp u6 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4)))) x1.
Apply H5 with
False.
Assume H7:
and (atleastp u6 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3)).
Apply H7 with
False.
Assume H9:
∀ x2 . ... ⟶ ∀ x3 . x3 ∈ ... ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3).