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Theorem iscard3 7736
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 7593 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2356 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 202 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 eloni 4418 . . . . . . . 8  |-  ( A  e.  On  ->  Ord  A )
53, 4syl 15 . . . . . . 7  |-  ( (
card `  A )  =  A  ->  Ord  A
)
6 ordom 4681 . . . . . . 7  |-  Ord  om
7 ordtri2or 4504 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  om )  ->  ( A  e.  om  \/  om  C_  A
) )
85, 6, 7sylancl 643 . . . . . 6  |-  ( (
card `  A )  =  A  ->  ( A  e.  om  \/  om  C_  A ) )
98ord 366 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( -.  A  e.  om  ->  om  C_  A ) )
10 isinfcard 7735 . . . . . . 7  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
1110biimpi 186 . . . . . 6  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  e.  ran  aleph )
1211expcom 424 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( om  C_  A  ->  A  e. 
ran  aleph ) )
139, 12syld 40 . . . 4  |-  ( (
card `  A )  =  A  ->  ( -.  A  e.  om  ->  A  e.  ran  aleph ) )
1413orrd 367 . . 3  |-  ( (
card `  A )  =  A  ->  ( A  e.  om  \/  A  e.  ran  aleph ) )
15 cardnn 7612 . . . 4  |-  ( A  e.  om  ->  ( card `  A )  =  A )
1610bicomi 193 . . . . 5  |-  ( A  e.  ran  aleph  <->  ( om  C_  A  /\  ( card `  A )  =  A ) )
1716simprbi 450 . . . 4  |-  ( A  e.  ran  aleph  ->  ( card `  A )  =  A )
1815, 17jaoi 368 . . 3  |-  ( ( A  e.  om  \/  A  e.  ran  aleph )  -> 
( card `  A )  =  A )
1914, 18impbii 180 . 2  |-  ( (
card `  A )  =  A  <->  ( A  e. 
om  \/  A  e.  ran  aleph ) )
20 elun 3329 . 2  |-  ( A  e.  ( om  u.  ran  aleph )  <->  ( A  e.  om  \/  A  e. 
ran  aleph ) )
2119, 20bitr4i 243 1  |-  ( (
card `  A )  =  A  <->  A  e.  ( om  u.  ran  aleph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   Ord word 4407   Oncon0 4408   omcom 4672   ran crn 4706   ` cfv 5271   cardccrd 7584   alephcale 7585
This theorem is referenced by:  cardnum  7737  carduniima  7739  cardinfima  7740  cfpwsdom  8222  gch2  8317
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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