Theorem oaord1

Description: An ordinal is less than its sum with a nonzero ordinal. Theorem 18 of Suppes p. 209 and its converse. (Contributed by NM, 6-Dec-2004)

	Ref	Expression
Assertion	oaord1	|- ( ( A e. On /\ B e. On ) -> ( (/) e. B <-> A e. ( A +o B ) ) )

Proof

Step	Hyp	Ref	Expression
1		0elon	|- (/) e. On
2		oaord	|- ( ( (/) e. On /\ B e. On /\ A e. On ) -> ( (/) e. B <-> ( A +o (/) ) e. ( A +o B ) ) )
3	1 2	mp3an1	|- ( ( B e. On /\ A e. On ) -> ( (/) e. B <-> ( A +o (/) ) e. ( A +o B ) ) )
4		oa0	|- ( A e. On -> ( A +o (/) ) = A )
5	4	adantl	|- ( ( B e. On /\ A e. On ) -> ( A +o (/) ) = A )
6	5	eleq1d	|- ( ( B e. On /\ A e. On ) -> ( ( A +o (/) ) e. ( A +o B ) <-> A e. ( A +o B ) ) )
7	3 6	bitrd	|- ( ( B e. On /\ A e. On ) -> ( (/) e. B <-> A e. ( A +o B ) ) )
8	7	ancoms	|- ( ( A e. On /\ B e. On ) -> ( (/) e. B <-> A e. ( A +o B ) ) )