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Theorem infenaleph 7905
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 7779 . . . . 5  |-  ( card `  ( card `  A
) )  =  (
card `  A )
2 cardom 7806 . . . . . . 7  |-  ( card `  om )  =  om
3 simpr 448 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  A )
4 omelon 7534 . . . . . . . . . 10  |-  om  e.  On
5 onenon 7769 . . . . . . . . . 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 7797 . . . . . . . . 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 3336 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om  C_  ( card `  A
) )
12 cardalephex 7904 . . . . . 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 2389 . . . . 5  |-  ( (
card `  A )  =  ( aleph `  x
)  <->  ( aleph `  x
)  =  ( card `  A ) )
1615rexbii 2674 . . . 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 7879 . . . 4  |-  aleph  Fn  On
19 fvelrnb 5713 . . . 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 7773 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2322adantr 452 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  ~~  A )
24 breq1 4156 . . 3  |-  ( x  =  ( card `  A
)  ->  ( x  ~~  A  <->  ( card `  A
)  ~~  A )
)
2524rspcev 2995 . 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 1717   E.wrex 2650    C_ wss 3263   class class class wbr 4153   Oncon0 4522   omcom 4785   dom cdm 4818   ran crn 4819    Fn wfn 5389   ` cfv 5394    ~~ cen 7042    ~<_ cdom 7043   cardccrd 7755   alephcale 7756
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-oi 7412  df-har 7459  df-card 7759  df-aleph 7760
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