MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isinfcard Structured version   Unicode version

Theorem isinfcard 7973
Description: Two ways to express the property of being a transfinite cardinal. (Contributed by NM, 9-Nov-2003.)
Assertion
Ref Expression
isinfcard  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )

Proof of Theorem isinfcard
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 alephfnon 7946 . . 3  |-  aleph  Fn  On
2 fvelrnb 5774 . . 3  |-  ( aleph  Fn  On  ->  ( A  e.  ran  aleph 
<->  E. x  e.  On  ( aleph `  x )  =  A ) )
31, 2ax-mp 8 . 2  |-  ( A  e.  ran  aleph  <->  E. x  e.  On  ( aleph `  x
)  =  A )
4 alephgeom 7963 . . . . . . 7  |-  ( x  e.  On  <->  om  C_  ( aleph `  x ) )
54biimpi 187 . . . . . 6  |-  ( x  e.  On  ->  om  C_  ( aleph `  x ) )
6 sseq2 3370 . . . . . 6  |-  ( A  =  ( aleph `  x
)  ->  ( om  C_  A  <->  om  C_  ( aleph `  x ) ) )
75, 6syl5ibrcom 214 . . . . 5  |-  ( x  e.  On  ->  ( A  =  ( aleph `  x )  ->  om  C_  A
) )
87rexlimiv 2824 . . . 4  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  ->  om  C_  A
)
98pm4.71ri 615 . . 3  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  <->  ( om  C_  A  /\  E. x  e.  On  A  =  ( aleph `  x ) ) )
10 eqcom 2438 . . . 4  |-  ( (
aleph `  x )  =  A  <->  A  =  ( aleph `  x ) )
1110rexbii 2730 . . 3  |-  ( E. x  e.  On  ( aleph `  x )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) )
12 cardalephex 7971 . . . 4  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
1312pm5.32i 619 . . 3  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  ( om  C_  A  /\  E. x  e.  On  A  =  (
aleph `  x ) ) )
149, 11, 133bitr4i 269 . 2  |-  ( E. x  e.  On  ( aleph `  x )  =  A  <->  ( om  C_  A  /\  ( card `  A
)  =  A ) )
153, 14bitr2i 242 1  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    C_ wss 3320   Oncon0 4581   omcom 4845   ran crn 4879    Fn wfn 5449   ` cfv 5454   cardccrd 7822   alephcale 7823
This theorem is referenced by:  iscard3  7974  alephinit  7976  cardinfima  7978  alephiso  7979  alephsson  7981  alephfp  7989
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
  Copyright terms: Public domain W3C validator