Let x0 of type ι be given.

Let x1 of type ι → ο be given.

Let x2 of type ι → ο be given.

Assume H0: PNoLt_ x0 x1 x2.

Let x3 of type ο be given.

Assume H1: ∀ x4 . x4 ∈ x0 ⟶ PNoEq_ x4 x1 x2 ⟶ not (x1 x4) ⟶ x2 x4 ⟶ x3.

Apply H0 with x3.

Let x4 of type ι be given.

Assume H2: (λ x5 . and (x5 ∈ x0) (and (and (PNoEq_ x5 x1 x2) (not (x1 x5))) (x2 x5))) x4.

Apply H2 with x3.

Assume H3: x4 ∈ x0.

Assume H4: and (and (PNoEq_ x4 x1 x2) (not (x1 x4))) (x2 x4).

Apply H4 with x3.

Assume H5: and (PNoEq_ x4 x1 x2) (not (x1 x4)).

Apply H5 with x2 x4 ⟶ x3.

Assume H6: PNoEq_ x4 x1 x2.

Assume H7: not (x1 x4).

Assume H8: x2 x4.

Apply H1 with x4 leaving 4 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H6.

The subproof is completed by applying H7.

The subproof is completed by applying H8.

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Proofgold Proof