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Theorem iscard3 7974
Description: Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.)
Assertion
Ref Expression
iscard3  |-  ( (
card `  A )  =  A  <->  A  e.  ( om  u.  ran  aleph ) )

Proof of Theorem iscard3
StepHypRef Expression
1 cardon 7831 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2496 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 203 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 eloni 4591 . . . . . . . 8  |-  ( A  e.  On  ->  Ord  A )
53, 4syl 16 . . . . . . 7  |-  ( (
card `  A )  =  A  ->  Ord  A
)
6 ordom 4854 . . . . . . 7  |-  Ord  om
7 ordtri2or 4677 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  om )  ->  ( A  e.  om  \/  om  C_  A
) )
85, 6, 7sylancl 644 . . . . . 6  |-  ( (
card `  A )  =  A  ->  ( A  e.  om  \/  om  C_  A ) )
98ord 367 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( -.  A  e.  om  ->  om  C_  A ) )
10 isinfcard 7973 . . . . . . 7  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
1110biimpi 187 . . . . . 6  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  e.  ran  aleph )
1211expcom 425 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( om  C_  A  ->  A  e. 
ran  aleph ) )
139, 12syld 42 . . . 4  |-  ( (
card `  A )  =  A  ->  ( -.  A  e.  om  ->  A  e.  ran  aleph ) )
1413orrd 368 . . 3  |-  ( (
card `  A )  =  A  ->  ( A  e.  om  \/  A  e.  ran  aleph ) )
15 cardnn 7850 . . . 4  |-  ( A  e.  om  ->  ( card `  A )  =  A )
1610bicomi 194 . . . . 5  |-  ( A  e.  ran  aleph  <->  ( om  C_  A  /\  ( card `  A )  =  A ) )
1716simprbi 451 . . . 4  |-  ( A  e.  ran  aleph  ->  ( card `  A )  =  A )
1815, 17jaoi 369 . . 3  |-  ( ( A  e.  om  \/  A  e.  ran  aleph )  -> 
( card `  A )  =  A )
1914, 18impbii 181 . 2  |-  ( (
card `  A )  =  A  <->  ( A  e. 
om  \/  A  e.  ran  aleph ) )
20 elun 3488 . 2  |-  ( A  e.  ( om  u.  ran  aleph )  <->  ( A  e.  om  \/  A  e. 
ran  aleph ) )
2119, 20bitr4i 244 1  |-  ( (
card `  A )  =  A  <->  A  e.  ( om  u.  ran  aleph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3318    C_ wss 3320   Ord word 4580   Oncon0 4581   omcom 4845   ran crn 4879   ` cfv 5454   cardccrd 7822   alephcale 7823
This theorem is referenced by:  cardnum  7975  carduniima  7977  cardinfima  7978  cfpwsdom  8459  gch2  8554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-har 7526  df-card 7826  df-aleph 7827
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