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Theorem isinfcard 8464
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 8437 . . 3  |-  aleph  Fn  On
2 fvelrnb 5895 . . 3  |-  ( aleph  Fn  On  ->  ( A  e.  ran  aleph 
<->  E. x  e.  On  ( aleph `  x )  =  A ) )
31, 2ax-mp 5 . 2  |-  ( A  e.  ran  aleph  <->  E. x  e.  On  ( aleph `  x
)  =  A )
4 alephgeom 8454 . . . . . . 7  |-  ( x  e.  On  <->  om  C_  ( aleph `  x ) )
54biimpi 194 . . . . . 6  |-  ( x  e.  On  ->  om  C_  ( aleph `  x ) )
6 sseq2 3511 . . . . . 6  |-  ( A  =  ( aleph `  x
)  ->  ( om  C_  A  <->  om  C_  ( aleph `  x ) ) )
75, 6syl5ibrcom 222 . . . . 5  |-  ( x  e.  On  ->  ( A  =  ( aleph `  x )  ->  om  C_  A
) )
87rexlimiv 2940 . . . 4  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  ->  om  C_  A
)
98pm4.71ri 631 . . 3  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  <->  ( om  C_  A  /\  E. x  e.  On  A  =  ( aleph `  x ) ) )
10 eqcom 2463 . . . 4  |-  ( (
aleph `  x )  =  A  <->  A  =  ( aleph `  x ) )
1110rexbii 2956 . . 3  |-  ( E. x  e.  On  ( aleph `  x )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) )
12 cardalephex 8462 . . . 4  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
1312pm5.32i 635 . . 3  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  ( om  C_  A  /\  E. x  e.  On  A  =  (
aleph `  x ) ) )
149, 11, 133bitr4i 277 . 2  |-  ( E. x  e.  On  ( aleph `  x )  =  A  <->  ( om  C_  A  /\  ( card `  A
)  =  A ) )
153, 14bitr2i 250 1  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805    C_ wss 3461   Oncon0 4867   ran crn 4989    Fn wfn 5565   ` cfv 5570   omcom 6673   cardccrd 8307   alephcale 8308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-har 7976  df-card 8311  df-aleph 8312
This theorem is referenced by:  iscard3  8465  alephinit  8467  cardinfima  8469  alephiso  8470  alephsson  8472  alephfp  8480
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