Theorem iscard 8434

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 8403	. . 3 |- ( card ` A ) e. On
2		eleq1 2527	. . 3 |- ( ( card ` A ) = A -> ( ( card ` A ) e. On <-> A e. On ) )
3	1, 2	mpbii 216	. 2 |- ( ( card ` A ) = A -> A e. On )
4		cardonle 8416	. . . 4 |- ( A e. On -> ( card ` A ) C_ A )
5		eqss 3458	. . . . 5 |- ( ( card ` A ) = A <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) )
6	5	baibr 920	. . . 4 |- ( ( card ` A ) C_ A -> ( A C_ ( card ` A ) <-> ( card ` A ) = A ) )
7	4, 6	syl 17	. . 3 |- ( A e. On -> ( A C_ ( card ` A ) <-> ( card ` A ) = A ) )
8		onelon 5466	. . . . . 6 |- ( ( A e. On /\ x e. A ) -> x e. On )
9		onenon 8408	. . . . . . 7 |- ( A e. On -> A e. dom card )
10	9	adantr 471	. . . . . 6 |- ( ( A e. On /\ x e. A ) -> A e. dom card )
11		cardsdomel 8433	. . . . . 6 |- ( ( x e. On /\ A e. dom card ) -> ( x ~< A <-> x e. ( card ` A ) ) )
12	8, 10, 11	syl2anc 671	. . . . 5 |- ( ( A e. On /\ x e. A ) -> ( x ~< A <-> x e. ( card ` A ) ) )
13	12	ralbidva 2835	. . . 4 |- ( A e. On -> ( A. x e. A x ~< A <-> A. x e. A x e. ( card ` A ) ) )
14		dfss3 3433	. . . 4 |- ( A C_ ( card ` A ) <-> A. x e. A x e. ( card ` A ) )
15	13, 14	syl6rbbr 272	. . 3 |- ( A e. On -> ( A C_ ( card ` A ) <-> A. x e. A x ~< A ) )
16	7, 15	bitr3d 263	. 2 |- ( A e. On -> ( ( card ` A ) = A <-> A. x e. A x ~< A ) )
17	3, 16	biadan2 652	1 |- ( ( card ` A ) = A <-> ( A e. On /\ A. x e. A x ~< A ) )

Syntax hints:   <-> wb 189   /\ wa 375   = wceq 1454   e. wcel 1897   A.wral 2748   C_ wss 3415   class class class wbr 4415   dom cdm 4852   Oncon0 5441   ` cfv 5600   ~< csdm 7593   cardccrd 8394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-ord 5444  df-on 5445  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-card 8398

This theorem is referenced by:  cardprclem  8438  cardmin2  8457  infxpenlem  8469  alephsuc2  8536  cardmin  9014  alephreg  9032  pwcfsdom  9033  winalim2  9146  gchina  9149  inar1  9225  r1tskina  9232  gruina  9268