Theorem iscard 8347

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 8316	. . 3 |- ( card ` A ) e. On
2		eleq1 2526	. . 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 8329	. . . 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 902	. . . 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 4892	. . . . . 6 |- ( ( A e. On /\ x e. A ) -> x e. On )
9		onenon 8321	. . . . . . 7 |- ( A e. On -> A e. dom card )
10	9	adantr 463	. . . . . 6 |- ( ( A e. On /\ x e. A ) -> A e. dom card )
11		cardsdomel 8346	. . . . . 6 |- ( ( x e. On /\ A e. dom card ) -> ( x ~< A <-> x e. ( card ` A ) ) )
12	8, 10, 11	syl2anc 659	. . . . 5 |- ( ( A e. On /\ x e. A ) -> ( x ~< A <-> x e. ( card ` A ) ) )
13	12	ralbidva 2890	. . . 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 640	1 |- ( ( card ` A ) = A <-> ( A e. On /\ A. x e. A x ~< A ) )

Syntax hints:   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   C_ wss 3461   class class class wbr 4439   Oncon0 4867   dom cdm 4988   ` cfv 5570   ~< csdm 7508   cardccrd 8307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-card 8311

This theorem is referenced by:  cardprclem  8351  cardmin2  8370  infxpenlem  8382  alephsuc2  8452  cardmin  8930  alephreg  8948  pwcfsdom  8949  winalim2  9063  gchina  9066  inar1  9142  r1tskina  9149  gruina  9185