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Theorem iscard 7608
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 7577 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2343 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 202 . 2  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 cardonle 7590 . . . 4  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
5 eqss 3194 . . . . 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 4417 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  A )  ->  x  e.  On )
9 onenon 7582 . . . . . . 7  |-  ( A  e.  On  ->  A  e.  dom  card )
109adantr 451 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  A )  ->  A  e.  dom  card )
11 cardsdomel 7607 . . . . . 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 2559 . . . 4  |-  ( A  e.  On  ->  ( A. x  e.  A  x  ~<  A  <->  A. x  e.  A  x  e.  ( card `  A )
) )
14 dfss3 3170 . . . 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 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~< csdm 6862   cardccrd 7568
This theorem is referenced by:  cardprclem  7612  cardmin2  7631  infxpenlem  7641  alephsuc2  7707  cardmin  8186  alephreg  8204  pwcfsdom  8205  winalim2  8318  gchina  8321  inar1  8397  r1tskina  8404  gruina  8440  carinttar  25902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572
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