Theorem iscard 8166

Description: Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)

Assertion

Ref	Expression
iscard	|- ( ( card ` A ) = A <-> ( A e. On /\ A. x e. A x ~< A ) )

Distinct variable group:   x, A

Proof of Theorem iscard

Step	Hyp	Ref	Expression
1		cardon 8135	. . 3 |- ( card ` A ) e. On
2		eleq1 2503	. . 3 |- ( ( card ` A ) = A -> ( ( card ` A ) e. On <-> A e. On ) )
3	1, 2	mpbii 211	. 2 |- ( ( card ` A ) = A -> A e. On )
4		cardonle 8148	. . . 4 |- ( A e. On -> ( card ` A ) C_ A )
5		eqss 3392	. . . . 5 |- ( ( card ` A ) = A <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) )
6	5	baibr 897	. . . 4 |- ( ( card ` A ) C_ A -> ( A C_ ( card ` A ) <-> ( card ` A ) = A ) )
7	4, 6	syl 16	. . 3 |- ( A e. On -> ( A C_ ( card ` A ) <-> ( card ` A ) = A ) )
8		onelon 4765	. . . . . 6 |- ( ( A e. On /\ x e. A ) -> x e. On )
9		onenon 8140	. . . . . . 7 |- ( A e. On -> A e. dom card )
10	9	adantr 465	. . . . . 6 |- ( ( A e. On /\ x e. A ) -> A e. dom card )
11		cardsdomel 8165	. . . . . 6 |- ( ( x e. On /\ A e. dom card ) -> ( x ~< A <-> x e. ( card ` A ) ) )
12	8, 10, 11	syl2anc 661	. . . . 5 |- ( ( A e. On /\ x e. A ) -> ( x ~< A <-> x e. ( card ` A ) ) )
13	12	ralbidva 2752	. . . 4 |- ( A e. On -> ( A. x e. A x ~< A <-> A. x e. A x e. ( card ` A ) ) )
14		dfss3 3367	. . . 4 |- ( A C_ ( card ` A ) <-> A. x e. A x e. ( card ` A ) )
15	13, 14	syl6rbbr 264	. . 3 |- ( A e. On -> ( A C_ ( card ` A ) <-> A. x e. A x ~< A ) )
16	7, 15	bitr3d 255	. 2 |- ( A e. On -> ( ( card ` A ) = A <-> A. x e. A x ~< A ) )
17	3, 16	biadan2 642	1 |- ( ( card ` A ) = A <-> ( A e. On /\ A. x e. A x ~< A ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2736   C_ wss 3349   class class class wbr 4313   Oncon0 4740   dom cdm 4861   ` cfv 5439   ~< csdm 7330   cardccrd 8126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-card 8130

This theorem is referenced by:  cardprclem  8170  cardmin2  8189  infxpenlem  8201  alephsuc2  8271  cardmin  8749  alephreg  8767  pwcfsdom  8768  winalim2  8884  gchina  8887  inar1  8963  r1tskina  8970  gruina  9006