Metamath Proof Explorer


Theorem ordsuc

Description: A class is ordinal if and only if its successor is ordinal. (Contributed by NM, 3-Apr-1995) Avoid ax-un . (Revised by BTernaryTau, 6-Jan-2025)

Ref Expression
Assertion ordsuc Ord A Ord suc A

Proof

Step Hyp Ref Expression
1 ordsuci Ord A Ord suc A
2 sucidg A V A suc A
3 ordelord Ord suc A A suc A Ord A
4 3 ex Ord suc A A suc A Ord A
5 2 4 syl5com A V Ord suc A Ord A
6 sucprc ¬ A V suc A = A
7 6 eqcomd ¬ A V A = suc A
8 ordeq A = suc A Ord A Ord suc A
9 7 8 syl ¬ A V Ord A Ord suc A
10 9 biimprd ¬ A V Ord suc A Ord A
11 5 10 pm2.61i Ord suc A Ord A
12 1 11 impbii Ord A Ord suc A