Theorem ordsssuc

Description: An ordinal is a subset of another ordinal if and only if it belongs to its successor. (Contributed by NM, 28-Nov-2003)

	Ref	Expression
Assertion	ordsssuc	|- ( ( A e. On /\ Ord B ) -> ( A C_ B <-> A e. suc B ) )

Proof

Step	Hyp	Ref	Expression
1		eloni	|- ( A e. On -> Ord A )
2		ordsseleq	|- ( ( Ord A /\ Ord B ) -> ( A C_ B <-> ( A e. B \/ A = B ) ) )
3	1 2	sylan	|- ( ( A e. On /\ Ord B ) -> ( A C_ B <-> ( A e. B \/ A = B ) ) )
4		elsucg	|- ( A e. On -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
5	4	adantr	|- ( ( A e. On /\ Ord B ) -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
6	3 5	bitr4d	|- ( ( A e. On /\ Ord B ) -> ( A C_ B <-> A e. suc B ) )