Theorem onsucb

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc . (Contributed by NM, 9-Sep-2003)

	Ref	Expression
Assertion	onsucb	|- ( A e. On <-> suc A e. On )

Proof

Step	Hyp	Ref	Expression
1		ordsuc	|- ( Ord A <-> Ord suc A )
2		sucexb	|- ( A e. _V <-> suc A e. _V )
3	1 2	anbi12i	|- ( ( Ord A /\ A e. _V ) <-> ( Ord suc A /\ suc A e. _V ) )
4		elon2	|- ( A e. On <-> ( Ord A /\ A e. _V ) )
5		elon2	|- ( suc A e. On <-> ( Ord suc A /\ suc A e. _V ) )
6	3 4 5	3bitr4i	|- ( A e. On <-> suc A e. On )