MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscard Structured version   Unicode version

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
StepHypRef Expression
1 cardon 8330 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2494 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 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 ) ) )
65baibr 912 . . . 4  |-  ( (
card `  A )  C_  A  ->  ( A  C_  ( card `  A
)  <->  ( card `  A
)  =  A ) )
74, 6syl 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 )
109adantr 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 )
) )
128, 10, 11syl2anc 665 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ( x  ~<  A  <->  x  e.  ( card `  A )
) )
1312ralbidva 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 ) )
1513, 14syl6rbbr 267 . . 3  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  A. x  e.  A  x  ~<  A ) )
167, 15bitr3d 258 . 2  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A. x  e.  A  x  ~<  A ) )
173, 16biadan2 646 1  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator