Theorem iscard 8361

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 8330	. . 3 |- ( card ` A ) e. On
2		eleq1 2494	. . 3 |- ( ( card ` A ) = A -> ( ( card ` A ) e. On <-> A e. On ) )
3	1, 2	mpbii 214	. 2 |- ( ( card ` A ) = A -> A e. On )
4		cardonle 8343	. . . 4 |- ( A e. On -> ( card ` A ) C_ A )
5		eqss 3422	. . . . 5 |- ( ( card ` A ) = A <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) )
6	5	baibr 912	. . . 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 5410	. . . . . 6 |- ( ( A e. On /\ x e. A ) -> x e. On )
9		onenon 8335	. . . . . . 7 |- ( A e. On -> A e. dom card )
10	9	adantr 466	. . . . . 6 |- ( ( A e. On /\ x e. A ) -> A e. dom card )
11		cardsdomel 8360	. . . . . 6 |- ( ( x e. On /\ A e. dom card ) -> ( x ~< A <-> x e. ( card ` A ) ) )
12	8, 10, 11	syl2anc 665	. . . . 5 |- ( ( A e. On /\ x e. A ) -> ( x ~< A <-> x e. ( card ` A ) ) )
13	12	ralbidva 2801	. . . 4 |- ( A e. On -> ( A. x e. A x ~< A <-> A. x e. A x e. ( card ` A ) ) )
14		dfss3 3397	. . . 4 |- ( A C_ ( card ` A ) <-> A. x e. A x e. ( card ` A ) )
15	13, 14	syl6rbbr 267	. . 3 |- ( A e. On -> ( A C_ ( card ` A ) <-> A. x e. A x ~< A ) )
16	7, 15	bitr3d 258	. 2 |- ( A e. On -> ( ( card ` A ) = A <-> A. x e. A x ~< A ) )
17	3, 16	biadan2 646	1 |- ( ( card ` A ) = A <-> ( A e. On /\ A. x e. A x ~< A ) )

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1872   A.wral 2714   C_ wss 3379   class class class wbr 4366   dom cdm 4796   Oncon0 5385   ` cfv 5544   ~< csdm 7523   cardccrd 8321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-card 8325

This theorem is referenced by:  cardprclem  8365  cardmin2  8384  infxpenlem  8396  alephsuc2  8462  cardmin  8940  alephreg  8958  pwcfsdom  8959  winalim2  9072  gchina  9075  inar1  9151  r1tskina  9158  gruina  9194