Let x0 of type ι be given.

Let x1 of type ι be given.

Assume H0: ordinal x0.

Assume H1: ordinal x1.

Let x2 of type ι → ο be given.

Let x3 of type ι → ο be given.

Let x4 of type ι → ο be given.

Assume H2: PNoEq_ x0 x2 x3.

Assume H3: or (40dde.. x0 x3 x1 x4) (and (x0 = x1) (PNoEq_ x0 x3 x4)).

Apply H3 with or (40dde.. x0 x2 x1 x4) (and (x0 = x1) (PNoEq_ x0 x2 x4)) leaving 2 subgoals.

Assume H4: 40dde.. x0 x3 x1 x4.

Apply orIL with 40dde.. x0 x2 x1 x4, and (x0 = x1) (PNoEq_ x0 x2 x4).

Apply unknownprop_516ec77a0547bdde87f302357c77a8c500e4737d03bed523bd783f1c87c1572b with x0, x1, x2, x3, x4 leaving 4 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

The subproof is completed by applying H4.

Assume H4: and (x0 = x1) (PNoEq_ x0 x3 x4).

Apply H4 with or (40dde.. x0 x2 x1 x4) (and (x0 = x1) (PNoEq_ x0 x2 x4)).

Assume H5: x0 = x1.

Assume H6: PNoEq_ x0 x3 x4.

Apply orIR with 40dde.. x0 x2 x1 x4, and (x0 = x1) (PNoEq_ x0 x2 x4).

Apply andI with x0 = x1, PNoEq_ x0 x2 x4 leaving 2 subgoals.

The subproof is completed by applying H5.

Apply PNoEq_tra_ with x0, x2, x3, x4 leaving 2 subgoals.

The subproof is completed by applying H2.

The subproof is completed by applying H6.

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Proofgold Proof