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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: ordinal x1.
Apply or3E with x0x1, x0 = x1, x1x0, exactly1of3 (x0x1) (x0 = x1) (x1x0) leaving 4 subgoals.
Apply ordinal_trichotomy_or with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H2: x0x1.
Apply exactly1of3_I1 with x0x1, x0 = x1, x1x0 leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3: x0 = x1.
Apply In_irref with x0.
Apply H3 with λ x2 x3 . x0x3.
The subproof is completed by applying H2.
Assume H3: x1x0.
Apply In_no2cycle with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Assume H2: x0 = x1.
Apply exactly1of3_I2 with x0x1, x0 = x1, x1x0 leaving 3 subgoals.
Apply H2 with λ x2 x3 . nIn x3 x1.
The subproof is completed by applying In_irref with x1.
The subproof is completed by applying H2.
Apply H2 with λ x2 x3 . nIn x1 x3.
The subproof is completed by applying In_irref with x1.
Assume H2: x1x0.
Apply exactly1of3_I3 with x0x1, x0 = x1, x1x0 leaving 3 subgoals.
Assume H3: x0x1.
Apply In_no2cycle with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
Assume H3: x0 = x1.
Apply In_irref with x0.
Apply H3 with λ x2 x3 . x3x0.
The subproof is completed by applying H2.
The subproof is completed by applying H2.