Theorem iscard 8359

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 8328	. . 3 |- ( card ` A ) e. On
2		eleq1 2515	. . 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 8341	. . . 4 |- ( A e. On -> ( card ` A ) C_ A )
5		eqss 3504	. . . . 5 |- ( ( card ` A ) = A <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) )
6	5	baibr 904	. . . 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 4893	. . . . . 6 |- ( ( A e. On /\ x e. A ) -> x e. On )
9		onenon 8333	. . . . . . 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 8358	. . . . . 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 2879	. . . 4 |- ( A e. On -> ( A. x e. A x ~< A <-> A. x e. A x e. ( card ` A ) ) )
14		dfss3 3479	. . . 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 1383   e. wcel 1804   A.wral 2793   C_ wss 3461   class class class wbr 4437   Oncon0 4868   dom cdm 4989   ` cfv 5578   ~< csdm 7517   cardccrd 8319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-card 8323

This theorem is referenced by:  cardprclem  8363  cardmin2  8382  infxpenlem  8394  alephsuc2  8464  cardmin  8942  alephreg  8960  pwcfsdom  8961  winalim2  9077  gchina  9080  inar1  9156  r1tskina  9163  gruina  9199