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Theorem isinfcard 8474
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 8447 . . 3  |-  aleph  Fn  On
2 fvelrnb 5915 . . 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 8464 . . . . . . 7  |-  ( x  e.  On  <->  om  C_  ( aleph `  x ) )
54biimpi 194 . . . . . 6  |-  ( x  e.  On  ->  om  C_  ( aleph `  x ) )
6 sseq2 3526 . . . . . 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 2949 . . . 4  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  ->  om  C_  A
)
98pm4.71ri 633 . . 3  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  <->  ( om  C_  A  /\  E. x  e.  On  A  =  ( aleph `  x ) ) )
10 eqcom 2476 . . . 4  |-  ( (
aleph `  x )  =  A  <->  A  =  ( aleph `  x ) )
1110rexbii 2965 . . 3  |-  ( E. x  e.  On  ( aleph `  x )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) )
12 cardalephex 8472 . . . 4  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
1312pm5.32i 637 . . 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 369    = wceq 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   Oncon0 4878   ran crn 5000    Fn wfn 5583   ` cfv 5588   omcom 6685   cardccrd 8317   alephcale 8318
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 6577  ax-inf2 8059
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-om 6686  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-oi 7936  df-har 7985  df-card 8321  df-aleph 8322
This theorem is referenced by:  iscard3  8475  alephinit  8477  cardinfima  8479  alephiso  8480  alephsson  8482  alephfp  8490
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