This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )