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Theorem iscard3 8522
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 8377 . . . . . . . . 9  |-  ( card `  A )  e.  On
2 eleq1 2501 . . . . . . . . 9  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 214 . . . . . . . 8  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 eloni 5452 . . . . . . . 8  |-  ( A  e.  On  ->  Ord  A )
53, 4syl 17 . . . . . . 7  |-  ( (
card `  A )  =  A  ->  Ord  A
)
6 ordom 6715 . . . . . . 7  |-  Ord  om
7 ordtri2or 5537 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  om )  ->  ( A  e.  om  \/  om  C_  A
) )
85, 6, 7sylancl 666 . . . . . 6  |-  ( (
card `  A )  =  A  ->  ( A  e.  om  \/  om  C_  A ) )
98ord 378 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( -.  A  e.  om  ->  om  C_  A ) )
10 isinfcard 8521 . . . . . . 7  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
1110biimpi 197 . . . . . 6  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  e.  ran  aleph )
1211expcom 436 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( om  C_  A  ->  A  e. 
ran  aleph ) )
139, 12syld 45 . . . 4  |-  ( (
card `  A )  =  A  ->  ( -.  A  e.  om  ->  A  e.  ran  aleph ) )
1413orrd 379 . . 3  |-  ( (
card `  A )  =  A  ->  ( A  e.  om  \/  A  e.  ran  aleph ) )
15 cardnn 8396 . . . 4  |-  ( A  e.  om  ->  ( card `  A )  =  A )
1610bicomi 205 . . . . 5  |-  ( A  e.  ran  aleph  <->  ( om  C_  A  /\  ( card `  A )  =  A ) )
1716simprbi 465 . . . 4  |-  ( A  e.  ran  aleph  ->  ( card `  A )  =  A )
1815, 17jaoi 380 . . 3  |-  ( ( A  e.  om  \/  A  e.  ran  aleph )  -> 
( card `  A )  =  A )
1914, 18impbii 190 . 2  |-  ( (
card `  A )  =  A  <->  ( A  e. 
om  \/  A  e.  ran  aleph ) )
20 elun 3612 . 2  |-  ( A  e.  ( om  u.  ran  aleph )  <->  ( A  e.  om  \/  A  e. 
ran  aleph ) )
2119, 20bitr4i 255 1  |-  ( (
card `  A )  =  A  <->  A  e.  ( om  u.  ran  aleph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    u. cun 3440    C_ wss 3442   ran crn 4855   Ord word 5441   Oncon0 5442   ` cfv 5601   omcom 6706   cardccrd 8368   alephcale 8369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-oi 8025  df-har 8073  df-card 8372  df-aleph 8373
This theorem is referenced by:  cardnum  8523  carduniima  8525  cardinfima  8526  cfpwsdom  9007  gch2  9099
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