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Theorem iscard3 8473
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 8324 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2539 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 211 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 eloni 4888 . . . . . . . 8  |-  ( A  e.  On  ->  Ord  A )
53, 4syl 16 . . . . . . 7  |-  ( (
card `  A )  =  A  ->  Ord  A
)
6 ordom 6688 . . . . . . 7  |-  Ord  om
7 ordtri2or 4973 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  om )  ->  ( A  e.  om  \/  om  C_  A
) )
85, 6, 7sylancl 662 . . . . . 6  |-  ( (
card `  A )  =  A  ->  ( A  e.  om  \/  om  C_  A ) )
98ord 377 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( -.  A  e.  om  ->  om  C_  A ) )
10 isinfcard 8472 . . . . . . 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 8343 . . . 4  |-  ( A  e.  om  ->  ( card `  A )  =  A )
1610bicomi 202 . . . . 5  |-  ( A  e.  ran  aleph  <->  ( om  C_  A  /\  ( card `  A )  =  A ) )
1716simprbi 464 . . . 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 3645 . 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 1379    e. wcel 1767    u. cun 3474    C_ wss 3476   Ord word 4877   Oncon0 4878   ran crn 5000   ` cfv 5587   omcom 6679   cardccrd 8315   alephcale 8316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-om 6680  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7934  df-har 7983  df-card 8319  df-aleph 8320
This theorem is referenced by:  cardnum  8474  carduniima  8476  cardinfima  8477  cfpwsdom  8958  gch2  9052
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