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

Theorem isinfcard 7735
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 7708 . . 3  |-  aleph  Fn  On
2 fvelrnb 5586 . . 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 7725 . . . . . . 7  |-  ( x  e.  On  <->  om  C_  ( aleph `  x ) )
54biimpi 186 . . . . . 6  |-  ( x  e.  On  ->  om  C_  ( aleph `  x ) )
6 sseq2 3213 . . . . . 6  |-  ( A  =  ( aleph `  x
)  ->  ( om  C_  A  <->  om  C_  ( aleph `  x ) ) )
75, 6syl5ibrcom 213 . . . . 5  |-  ( x  e.  On  ->  ( A  =  ( aleph `  x )  ->  om  C_  A
) )
87rexlimiv 2674 . . . 4  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  ->  om  C_  A
)
98pm4.71ri 614 . . 3  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  <->  ( om  C_  A  /\  E. x  e.  On  A  =  ( aleph `  x ) ) )
10 eqcom 2298 . . . 4  |-  ( (
aleph `  x )  =  A  <->  A  =  ( aleph `  x ) )
1110rexbii 2581 . . 3  |-  ( E. x  e.  On  ( aleph `  x )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) )
12 cardalephex 7733 . . . 4  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
1312pm5.32i 618 . . 3  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  ( om  C_  A  /\  E. x  e.  On  A  =  (
aleph `  x ) ) )
149, 11, 133bitr4i 268 . 2  |-  ( E. x  e.  On  ( aleph `  x )  =  A  <->  ( om  C_  A  /\  ( card `  A
)  =  A ) )
153, 14bitr2i 241 1  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   Oncon0 4408   omcom 4672   ran crn 4706    Fn wfn 5266   ` cfv 5271   cardccrd 7584   alephcale 7585
This theorem is referenced by:  iscard3  7736  alephinit  7738  cardinfima  7740  alephiso  7741  alephsson  7743  alephfp  7751
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
  Copyright terms: Public domain W3C validator