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

Theorem infenaleph 7928
Description: An infinite numerable set is equinumerous to an infinite initial ordinal. (Contributed by Jeff Hankins, 23-Oct-2009.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infenaleph  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  E. x  e.  ran  aleph
x  ~~  A )
Distinct variable group:    x, A

Proof of Theorem infenaleph
StepHypRef Expression
1 cardidm 7802 . . . . 5  |-  ( card `  ( card `  A
) )  =  (
card `  A )
2 cardom 7829 . . . . . . 7  |-  ( card `  om )  =  om
3 simpr 448 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  A )
4 omelon 7557 . . . . . . . . . 10  |-  om  e.  On
5 onenon 7792 . . . . . . . . . 10  |-  ( om  e.  On  ->  om  e.  dom  card )
64, 5ax-mp 8 . . . . . . . . 9  |-  om  e.  dom  card
7 simpl 444 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  A  e.  dom  card )
8 carddom2 7820 . . . . . . . . 9  |-  ( ( om  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  om )  C_  ( card `  A )  <->  om  ~<_  A ) )
96, 7, 8sylancr 645 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( card `  om )  C_  ( card `  A
)  <->  om  ~<_  A ) )
103, 9mpbird 224 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  om )  C_  ( card `  A )
)
112, 10syl5eqssr 3353 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om  C_  ( card `  A
) )
12 cardalephex 7927 . . . . . 6  |-  ( om  C_  ( card `  A
)  ->  ( ( card `  ( card `  A
) )  =  (
card `  A )  <->  E. x  e.  On  ( card `  A )  =  ( aleph `  x )
) )
1311, 12syl 16 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( card `  ( card `  A ) )  =  ( card `  A
)  <->  E. x  e.  On  ( card `  A )  =  ( aleph `  x
) ) )
141, 13mpbii 203 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  E. x  e.  On  ( card `  A )  =  ( aleph `  x
) )
15 eqcom 2406 . . . . 5  |-  ( (
card `  A )  =  ( aleph `  x
)  <->  ( aleph `  x
)  =  ( card `  A ) )
1615rexbii 2691 . . . 4  |-  ( E. x  e.  On  ( card `  A )  =  ( aleph `  x )  <->  E. x  e.  On  ( aleph `  x )  =  ( card `  A
) )
1714, 16sylib 189 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  E. x  e.  On  ( aleph `  x )  =  ( card `  A
) )
18 alephfnon 7902 . . . 4  |-  aleph  Fn  On
19 fvelrnb 5733 . . . 4  |-  ( aleph  Fn  On  ->  ( ( card `  A )  e. 
ran  aleph 
<->  E. x  e.  On  ( aleph `  x )  =  ( card `  A
) ) )
2018, 19ax-mp 8 . . 3  |-  ( (
card `  A )  e.  ran  aleph 
<->  E. x  e.  On  ( aleph `  x )  =  ( card `  A
) )
2117, 20sylibr 204 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  e.  ran  aleph )
22 cardid2 7796 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2322adantr 452 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  ~~  A )
24 breq1 4175 . . 3  |-  ( x  =  ( card `  A
)  ->  ( x  ~~  A  <->  ( card `  A
)  ~~  A )
)
2524rspcev 3012 . 2  |-  ( ( ( card `  A
)  e.  ran  aleph  /\  ( card `  A )  ~~  A )  ->  E. x  e.  ran  aleph x  ~~  A
)
2621, 23, 25syl2anc 643 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  E. x  e.  ran  aleph
x  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667    C_ wss 3280   class class class wbr 4172   Oncon0 4541   omcom 4804   dom cdm 4837   ran crn 4838    Fn wfn 5408   ` cfv 5413    ~~ cen 7065    ~<_ cdom 7066   cardccrd 7778   alephcale 7779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783
  Copyright terms: Public domain W3C validator