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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: ordinal x0.
Assume H1: x1x0.
Claim L2: ordinal (ordsucc x1)
Apply ordinal_ordsucc with x1.
Apply ordinal_Hered with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply ordinal_trichotomy_or with x0, ordsucc x1, or (ordsucc x1x0) (x0 = ordsucc x1) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L2.
Assume H3: or (x0ordsucc x1) (x0 = ordsucc x1).
Apply H3 with or (ordsucc x1x0) (x0 = ordsucc x1) leaving 2 subgoals.
Assume H4: x0ordsucc x1.
Apply FalseE with or (ordsucc x1x0) (x0 = ordsucc x1).
Apply ordsuccE with x1, x0, False leaving 3 subgoals.
The subproof is completed by applying H4.
Assume H5: x0x1.
Apply In_no2cycle with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
Assume H5: x0 = x1.
Apply In_irref with x0.
Apply H5 with λ x2 x3 . x3x0.
The subproof is completed by applying H1.
Assume H4: x0 = ordsucc x1.
Apply orIR with ordsucc x1x0, x0 = ordsucc x1.
The subproof is completed by applying H4.
Assume H3: ordsucc x1x0.
Apply orIL with ordsucc x1x0, x0 = ordsucc x1.
The subproof is completed by applying H3.