Theorem oneqmini 5474

Description: A way to show that an ordinal number equals the minimum of a 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
oneqmini	|- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A = |^| B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem oneqmini

Step	Hyp	Ref	Expression
1		ssint 4250	. . . . . 6 |- ( A C_ |^| B <-> A. x e. B A C_ x )
2		ssel 3426	. . . . . . . . . . . 12 |- ( B C_ On -> ( A e. B -> A e. On ) )
3		ssel 3426	. . . . . . . . . . . 12 |- ( B C_ On -> ( x e. B -> x e. On ) )
4	2, 3	anim12d 566	. . . . . . . . . . 11 |- ( B C_ On -> ( ( A e. B /\ x e. B ) -> ( A e. On /\ x e. On ) ) )
5		ontri1 5457	. . . . . . . . . . 11 |- ( ( A e. On /\ x e. On ) -> ( A C_ x <-> -. x e. A ) )
6	4, 5	syl6 34	. . . . . . . . . 10 |- ( B C_ On -> ( ( A e. B /\ x e. B ) -> ( A C_ x <-> -. x e. A ) ) )
7	6	expdimp 439	. . . . . . . . 9 |- ( ( B C_ On /\ A e. B ) -> ( x e. B -> ( A C_ x <-> -. x e. A ) ) )
8	7	pm5.74d 251	. . . . . . . 8 |- ( ( B C_ On /\ A e. B ) -> ( ( x e. B -> A C_ x ) <-> ( x e. B -> -. x e. A ) ) )
9		con2b 336	. . . . . . . 8 |- ( ( x e. B -> -. x e. A ) <-> ( x e. A -> -. x e. B ) )
10	8, 9	syl6bb 265	. . . . . . 7 |- ( ( B C_ On /\ A e. B ) -> ( ( x e. B -> A C_ x ) <-> ( x e. A -> -. x e. B ) ) )
11	10	ralbidv2 2823	. . . . . 6 |- ( ( B C_ On /\ A e. B ) -> ( A. x e. B A C_ x <-> A. x e. A -. x e. B ) )
12	1, 11	syl5bb 261	. . . . 5 |- ( ( B C_ On /\ A e. B ) -> ( A C_ |^| B <-> A. x e. A -. x e. B ) )
13	12	biimprd 227	. . . 4 |- ( ( B C_ On /\ A e. B ) -> ( A. x e. A -. x e. B -> A C_ |^| B ) )
14	13	expimpd 608	. . 3 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A C_ |^| B ) )
15		intss1 4249	. . . . 5 |- ( A e. B -> |^| B C_ A )
16	15	a1i 11	. . . 4 |- ( B C_ On -> ( A e. B -> |^| B C_ A ) )
17	16	adantrd 470	. . 3 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> |^| B C_ A ) )
18	14, 17	jcad 536	. 2 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> ( A C_ |^| B /\ |^| B C_ A ) ) )
19		eqss 3447	. 2 |- ( A = |^| B <-> ( A C_ |^| B /\ |^| B C_ A ) )
20	18, 19	syl6ibr 231	1 |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B ) -> A = |^| B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   C_ wss 3404   |^|cint 4234   Oncon0 5423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-on 5427

This theorem is referenced by:  oneqmin  6632  alephval3  8541  cfsuc  8687  alephval2  8997