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Theorem iscard 7624
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 7593 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2356 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 202 . 2  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 cardonle 7606 . . . 4  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
5 eqss 3207 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
65baibr 872 . . . 4  |-  ( (
card `  A )  C_  A  ->  ( A  C_  ( card `  A
)  <->  ( card `  A
)  =  A ) )
74, 6syl 15 . . 3  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  ( card `  A
)  =  A ) )
8 onelon 4433 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  A )  ->  x  e.  On )
9 onenon 7598 . . . . . . 7  |-  ( A  e.  On  ->  A  e.  dom  card )
109adantr 451 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  A )  ->  A  e.  dom  card )
11 cardsdomel 7623 . . . . . 6  |-  ( ( x  e.  On  /\  A  e.  dom  card )  ->  ( x  ~<  A  <->  x  e.  ( card `  A )
) )
128, 10, 11syl2anc 642 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ( x  ~<  A  <->  x  e.  ( card `  A )
) )
1312ralbidva 2572 . . . 4  |-  ( A  e.  On  ->  ( A. x  e.  A  x  ~<  A  <->  A. x  e.  A  x  e.  ( card `  A )
) )
14 dfss3 3183 . . . 4  |-  ( A 
C_  ( card `  A
)  <->  A. x  e.  A  x  e.  ( card `  A ) )
1513, 14syl6rbbr 255 . . 3  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  A. x  e.  A  x  ~<  A ) )
167, 15bitr3d 246 . 2  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A. x  e.  A  x  ~<  A ) )
173, 16biadan2 623 1  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039   Oncon0 4408   dom cdm 4705   ` cfv 5271    ~< csdm 6878   cardccrd 7584
This theorem is referenced by:  cardprclem  7628  cardmin2  7647  infxpenlem  7657  alephsuc2  7723  cardmin  8202  alephreg  8220  pwcfsdom  8221  winalim2  8334  gchina  8337  inar1  8413  r1tskina  8420  gruina  8456  carinttar  26005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588
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