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Theorem infenaleph 7964
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 7838 . . . . 5  |-  ( card `  ( card `  A
) )  =  (
card `  A )
2 cardom 7865 . . . . . . 7  |-  ( card `  om )  =  om
3 simpr 448 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  A )
4 omelon 7593 . . . . . . . . . 10  |-  om  e.  On
5 onenon 7828 . . . . . . . . . 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 7856 . . . . . . . . 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 3385 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om  C_  ( card `  A
) )
12 cardalephex 7963 . . . . . 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 2437 . . . . 5  |-  ( (
card `  A )  =  ( aleph `  x
)  <->  ( aleph `  x
)  =  ( card `  A ) )
1615rexbii 2722 . . . 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 7938 . . . 4  |-  aleph  Fn  On
19 fvelrnb 5766 . . . 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 7832 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2322adantr 452 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  ~~  A )
24 breq1 4207 . . 3  |-  ( x  =  ( card `  A
)  ->  ( x  ~~  A  <->  ( card `  A
)  ~~  A )
)
2524rspcev 3044 . 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 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   class class class wbr 4204   Oncon0 4573   omcom 4837   dom cdm 4870   ran crn 4871    Fn wfn 5441   ` cfv 5446    ~~ cen 7098    ~<_ cdom 7099   cardccrd 7814   alephcale 7815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-har 7518  df-card 7818  df-aleph 7819
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