Let x0 of type ι → ο be given.

Let x1 of type ι → ο be given.

Let x2 of type ι be given.

Assume H0: ordinal x2.

Let x3 of type ι be given.

Assume H1: prim1 x3 x2.

Assume H2: PNoLt_ x3 x0 x1.

Apply H2 with PNoLt_ x2 x0 x1.

Let x4 of type ι be given.

Assume H3: (λ x5 . and (prim1 x5 x3) (and (and (PNoEq_ x5 x0 x1) (not (x0 x5))) (x1 x5))) x4.

Apply H3 with PNoLt_ x2 x0 x1.

Assume H4: prim1 x4 x3.

Assume H5: and (and (PNoEq_ x4 x0 x1) (not (x0 x4))) (x1 x4).

Let x5 of type ο be given.

Assume H6: ∀ x6 . and (prim1 x6 x2) (and (and (PNoEq_ x6 x0 x1) (not (x0 x6))) (x1 x6)) ⟶ x5.

Apply H6 with x4.

Apply andI with prim1 x4 x2, and (and (PNoEq_ x4 x0 x1) (not (x0 x4))) (x1 x4) leaving 2 subgoals.

Apply H0 with prim1 x4 x2.

Assume H7: TransSet x2.

Assume H8: ∀ x6 . prim1 x6 x2 ⟶ TransSet x6.

Apply H7 with x3, x4 leaving 2 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H4.

The subproof is completed by applying H5.

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Proofgold Proof