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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: ordinal x0.
Let x1 of type ι be given.
Assume H1: ordinal x1.
Let x2 of type ι be given.
Assume H2: x2x0.
Claim L3: ordinal x2
Apply ordinal_Hered with x0, x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Claim L4: ordinal (add_SNo x0 x1)
Apply add_SNo_ordinal_ordinal with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Claim L5: ordinal (add_SNo x2 x1)
Apply add_SNo_ordinal_ordinal with x2, x1 leaving 2 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H1.
Apply ordinal_SNoLt_In with add_SNo x2 x1, add_SNo x0 x1 leaving 3 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying L4.
Apply add_SNo_Lt1 with x2, x1, x0 leaving 4 subgoals.
Apply ordinal_SNo with x2.
The subproof is completed by applying L3.
Apply ordinal_SNo with x1.
The subproof is completed by applying H1.
Apply ordinal_SNo with x0.
The subproof is completed by applying H0.
Apply ordinal_In_SNoLt with x0, x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.