Theorem oneqmin 7005

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.)

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem oneqmin

Step	Hyp	Ref	Expression
1		onint 6995	. . . 4 |- ( ( B C_ On /\ B =/= (/) ) -> |^| B e. B )
2		eleq1 2689	. . . 4 |- ( A = |^| B -> ( A e. B <-> |^| B e. B ) )
3	1, 2	syl5ibrcom 237	. . 3 |- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B -> A e. B ) )
4		eleq2 2690	. . . . . . 7 |- ( A = |^| B -> ( x e. A <-> x e. |^| B ) )
5	4	biimpd 219	. . . . . 6 |- ( A = |^| B -> ( x e. A -> x e. |^| B ) )
6		onnmin 7003	. . . . . . . 8 |- ( ( B C_ On /\ x e. B ) -> -. x e. |^| B )
7	6	ex 450	. . . . . . 7 |- ( B C_ On -> ( x e. B -> -. x e. |^| B ) )
8	7	con2d 129	. . . . . 6 |- ( B C_ On -> ( x e. |^| B -> -. x e. B ) )
9	5, 8	syl9r 78	. . . . 5 |- ( B C_ On -> ( A = |^| B -> ( x e. A -> -. x e. B ) ) )
10	9	ralrimdv 2968	. . . 4 |- ( B C_ On -> ( A = |^| B -> A. x e. A -. x e. B ) )
11	10	adantr 481	. . 3 |- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B -> A. x e. A -. x e. B ) )
12	3, 11	jcad 555	. 2 |- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B -> ( A e. B /\ A. x e. A -. x e. B ) ) )
13		oneqmini 5776	. . 3 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A = |^| B ) )
14	13	adantr 481	. 2 |- ( ( B C_ On /\ B =/= (/) ) -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A = |^| B ) )
15	12, 14	impbid 202	1 |- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B <-> ( A e. B /\ A. x e. A -. x e. B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   |^|cint 4475   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by: (None)