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Proofgold Proof

pf
Let x0 of type ι(ιο) → ο be given.
Let x1 of type ι be given.
Assume H0: ordinal x1.
Let x2 of type ιο be given.
Let x3 of type ιο be given.
Assume H1: PNoEq_ x1 x2 x3.
Assume H2: PNo_rel_strict_upperbd x0 x1 x2.
Let x4 of type ι be given.
Assume H3: x4x1.
Let x5 of type ιο be given.
Assume H4: PNo_downc x0 x4 x5.
Claim L5: ordinal x4
Apply ordinal_Hered with x1, x4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply PNoLtEq_tra with x4, x1, x5, x2, x3 leaving 4 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying H0.
Apply H2 with x4, x5 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H1.