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Theorem iscard3 8259
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 8110 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2501 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 211 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 eloni 4725 . . . . . . . 8  |-  ( A  e.  On  ->  Ord  A )
53, 4syl 16 . . . . . . 7  |-  ( (
card `  A )  =  A  ->  Ord  A
)
6 ordom 6484 . . . . . . 7  |-  Ord  om
7 ordtri2or 4810 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  om )  ->  ( A  e.  om  \/  om  C_  A
) )
85, 6, 7sylancl 657 . . . . . 6  |-  ( (
card `  A )  =  A  ->  ( A  e.  om  \/  om  C_  A ) )
98ord 377 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( -.  A  e.  om  ->  om  C_  A ) )
10 isinfcard 8258 . . . . . . 7  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
1110biimpi 194 . . . . . 6  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  e.  ran  aleph )
1211expcom 435 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( om  C_  A  ->  A  e. 
ran  aleph ) )
139, 12syld 44 . . . 4  |-  ( (
card `  A )  =  A  ->  ( -.  A  e.  om  ->  A  e.  ran  aleph ) )
1413orrd 378 . . 3  |-  ( (
card `  A )  =  A  ->  ( A  e.  om  \/  A  e.  ran  aleph ) )
15 cardnn 8129 . . . 4  |-  ( A  e.  om  ->  ( card `  A )  =  A )
1610bicomi 202 . . . . 5  |-  ( A  e.  ran  aleph  <->  ( om  C_  A  /\  ( card `  A )  =  A ) )
1716simprbi 461 . . . 4  |-  ( A  e.  ran  aleph  ->  ( card `  A )  =  A )
1815, 17jaoi 379 . . 3  |-  ( ( A  e.  om  \/  A  e.  ran  aleph )  -> 
( card `  A )  =  A )
1914, 18impbii 188 . 2  |-  ( (
card `  A )  =  A  <->  ( A  e. 
om  \/  A  e.  ran  aleph ) )
20 elun 3494 . 2  |-  ( A  e.  ( om  u.  ran  aleph )  <->  ( A  e.  om  \/  A  e. 
ran  aleph ) )
2119, 20bitr4i 252 1  |-  ( (
card `  A )  =  A  <->  A  e.  ( om  u.  ran  aleph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761    u. cun 3323    C_ wss 3325   Ord word 4714   Oncon0 4715   ran crn 4837   ` cfv 5415   omcom 6475   cardccrd 8101   alephcale 8102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-oi 7720  df-har 7769  df-card 8105  df-aleph 8106
This theorem is referenced by:  cardnum  8260  carduniima  8262  cardinfima  8263  cfpwsdom  8744  gch2  8838
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