Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Assume H0: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Let x5 of type ι be given.
Assume H4: x5 ∈ x0.
Let x6 of type ο be given.
Assume H6:
(x2 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ not (x1 x2 x4) ⟶ not (x1 x2 x3) ⟶ not (x1 x4 x3) ⟶ not (x1 x2 x5) ⟶ x1 x4 x5 ⟶ x1 x3 x5 ⟶ x6.
Apply H5 with
x6.
Assume H7: x2 = x3 ⟶ ∀ x7 : ο . x7.
Assume H8: x2 = x4 ⟶ ∀ x7 : ο . x7.
Assume H9: x3 = x4 ⟶ ∀ x7 : ο . x7.
Assume H10: x2 = x5 ⟶ ∀ x7 : ο . x7.
Assume H11: x3 = x5 ⟶ ∀ x7 : ο . x7.
Assume H12: x4 = x5 ⟶ ∀ x7 : ο . x7.
Assume H13:
not (x1 x2 x3).
Assume H14:
not (x1 x2 x4).
Assume H15:
not (x1 x3 x4).
Assume H16:
not (x1 x2 x5).
Assume H17: x1 x3 x5.
Assume H18: x1 x4 x5.
Claim L19:
∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ not (x1 x7 x8) ⟶ not (x1 x8 x7)Let x7 of type ι be given.
Assume H19: x7 ∈ x0.
Let x8 of type ι be given.
Assume H20: x8 ∈ x0.
Assume H21:
not (x1 x7 x8).
Assume H22: x1 x8 x7.
Apply H21.
Apply H0 with
x8,
x7 leaving 3 subgoals.
The subproof is completed by applying H20.
The subproof is completed by applying H19.
The subproof is completed by applying H22.
Apply H6 leaving 12 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H7.
Apply neq_i_sym with
x3,
x4.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
The subproof is completed by applying H12.
The subproof is completed by applying H11.
The subproof is completed by applying H14.
The subproof is completed by applying H13.
Apply L19 with
x3,
x4 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H15.
The subproof is completed by applying H16.
The subproof is completed by applying H18.
The subproof is completed by applying H17.