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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: nat_p x0.
Assume H1: nat_p x1.
Assume H2: x0x1.
Apply ordinal_In_Or_Subq with x0, x1, ordsucc x0ordsucc x1 leaving 4 subgoals.
Apply nat_p_ordinal with x0.
The subproof is completed by applying H0.
Apply nat_p_ordinal with x1.
The subproof is completed by applying H1.
Assume H3: x0x1.
Apply nat_trans with ordsucc x1, ordsucc x0 leaving 2 subgoals.
Apply nat_ordsucc with x1.
The subproof is completed by applying H1.
Apply nat_ordsucc_in_ordsucc with x1, x0 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Assume H3: x1x0.
Claim L4: x0 = x1
Apply set_ext with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply L4 with λ x2 x3 . ordsucc x3ordsucc x1.
The subproof is completed by applying Subq_ref with ordsucc x1.