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:
(x5 = x2 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x4 ⟶ ∀ x7 : ο . x7) ⟶ not (x1 x5 x2) ⟶ not (x1 x5 x3) ⟶ not (x1 x2 x3) ⟶ not (x1 x5 x4) ⟶ not (x1 x2 x4) ⟶ not (x1 x3 x4) ⟶ x6.
Claim L7:
∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ not (x1 x7 x8) ⟶ not (x1 x8 x7)Let x7 of type ι be given.
Assume H7: x7 ∈ x0.
Let x8 of type ι be given.
Assume H8: x8 ∈ x0.
Assume H9:
not (x1 x7 x8).
Assume H10: x1 x8 x7.
Apply H9.
Apply H0 with
x8,
x7 leaving 3 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H7.
The subproof is completed by applying H10.
Apply H5 with
x6.
Assume H8: x2 = x3 ⟶ ∀ x7 : ο . x7.
Assume H9: x2 = x4 ⟶ ∀ x7 : ο . x7.
Assume H10: x3 = x4 ⟶ ∀ x7 : ο . x7.
Assume H11: x2 = x5 ⟶ ∀ x7 : ο . x7.
Assume H12: x3 = x5 ⟶ ∀ x7 : ο . x7.
Assume H13: x4 = x5 ⟶ ∀ x7 : ο . x7.
Assume H14:
not (x1 x2 x3).
Assume H15:
not (x1 x2 x4).
Assume H16:
not (x1 x3 x4).
Assume H17:
not (x1 x2 x5).
Assume H18:
not (x1 x3 x5).
Assume H19:
not (x1 x4 x5).
Apply H6 leaving 12 subgoals.
Apply neq_i_sym with
x2,
x5.
The subproof is completed by applying H11.
Apply neq_i_sym with
x3,
x5.
The subproof is completed by applying H12.
The subproof is completed by applying H8.
Apply neq_i_sym with
x4,
x5.
The subproof is completed by applying H13.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply L7 with
x2,
x5 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying H17.
Apply L7 with
x3,
x5 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
The subproof is completed by applying H18.
The subproof is completed by applying H14.
Apply L7 with
x4,
x5 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H19.
The subproof is completed by applying H15.
The subproof is completed by applying H16.