Theorem oneqmin

Description: A way to show that an ordinal number equals the minimum of a nonempty collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003)

	Ref	Expression
Assertion	oneqmin	|- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B <-> ( A e. B /\ A. x e. A -. x e. B ) ) )

Proof

Step	Hyp	Ref	Expression
1		onint	|- ( ( B C_ On /\ B =/= (/) ) -> |^| B e. B )
2		eleq1	|- ( A = |^| B -> ( A e. B <-> |^| B e. B ) )
3	1 2	syl5ibrcom	|- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B -> A e. B ) )
4		eleq2	|- ( A = |^| B -> ( x e. A <-> x e. |^| B ) )
5	4	biimpd	|- ( A = |^| B -> ( x e. A -> x e. |^| B ) )
6		onnmin	|- ( ( B C_ On /\ x e. B ) -> -. x e. |^| B )
7	6	ex	|- ( B C_ On -> ( x e. B -> -. x e. |^| B ) )
8	7	con2d	|- ( B C_ On -> ( x e. |^| B -> -. x e. B ) )
9	5 8	syl9r	|- ( B C_ On -> ( A = |^| B -> ( x e. A -> -. x e. B ) ) )
10	9	ralrimdv	|- ( B C_ On -> ( A = |^| B -> A. x e. A -. x e. B ) )
11	10	adantr	|- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B -> A. x e. A -. x e. B ) )
12	3 11	jcad	|- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B -> ( A e. B /\ A. x e. A -. x e. B ) ) )
13		oneqmini	|- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A = |^| B ) )
14	13	adantr	|- ( ( B C_ On /\ B =/= (/) ) -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A = |^| B ) )
15	12 14	impbid	|- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B <-> ( A e. B /\ A. x e. A -. x e. B ) ) )