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Theorem infenaleph 8253
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 8121 . . . . 5  |-  ( card `  ( card `  A
) )  =  (
card `  A )
2 cardom 8148 . . . . . . 7  |-  ( card `  om )  =  om
3 simpr 461 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  A )
4 omelon 7844 . . . . . . . . . 10  |-  om  e.  On
5 onenon 8111 . . . . . . . . . 10  |-  ( om  e.  On  ->  om  e.  dom  card )
64, 5ax-mp 5 . . . . . . . . 9  |-  om  e.  dom  card
7 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  A  e.  dom  card )
8 carddom2 8139 . . . . . . . . 9  |-  ( ( om  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  om )  C_  ( card `  A )  <->  om  ~<_  A ) )
96, 7, 8sylancr 663 . . . . . . . 8  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( card `  om )  C_  ( card `  A
)  <->  om  ~<_  A ) )
103, 9mpbird 232 . . . . . . 7  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  om )  C_  ( card `  A )
)
112, 10syl5eqssr 3396 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om  C_  ( card `  A
) )
12 cardalephex 8252 . . . . . 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 211 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  E. x  e.  On  ( card `  A )  =  ( aleph `  x
) )
15 eqcom 2440 . . . . 5  |-  ( (
card `  A )  =  ( aleph `  x
)  <->  ( aleph `  x
)  =  ( card `  A ) )
1615rexbii 2735 . . . 4  |-  ( E. x  e.  On  ( card `  A )  =  ( aleph `  x )  <->  E. x  e.  On  ( aleph `  x )  =  ( card `  A
) )
1714, 16sylib 196 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  E. x  e.  On  ( aleph `  x )  =  ( card `  A
) )
18 alephfnon 8227 . . . 4  |-  aleph  Fn  On
19 fvelrnb 5734 . . . 4  |-  ( aleph  Fn  On  ->  ( ( card `  A )  e. 
ran  aleph 
<->  E. x  e.  On  ( aleph `  x )  =  ( card `  A
) ) )
2018, 19ax-mp 5 . . 3  |-  ( (
card `  A )  e.  ran  aleph 
<->  E. x  e.  On  ( aleph `  x )  =  ( card `  A
) )
2117, 20sylibr 212 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  e.  ran  aleph )
22 cardid2 8115 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2322adantr 465 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  ~~  A )
24 breq1 4290 . . 3  |-  ( x  =  ( card `  A
)  ->  ( x  ~~  A  <->  ( card `  A
)  ~~  A )
)
2524rspcev 3068 . 2  |-  ( ( ( card `  A
)  e.  ran  aleph  /\  ( card `  A )  ~~  A )  ->  E. x  e.  ran  aleph x  ~~  A
)
2621, 23, 25syl2anc 661 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  E. x  e.  ran  aleph
x  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2711    C_ wss 3323   class class class wbr 4287   Oncon0 4714   dom cdm 4835   ran crn 4836    Fn wfn 5408   ` cfv 5413   omcom 6471    ~~ cen 7299    ~<_ cdom 7300   cardccrd 8097   alephcale 8098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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 6047  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-oi 7716  df-har 7765  df-card 8101  df-aleph 8102
This theorem is referenced by: (None)
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