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Theorem ween 8205
Description: A set is numerable iff it can be well-ordered. (Contributed by Mario Carneiro, 5-Jan-2013.)
Assertion
Ref Expression
ween  |-  ( A  e.  dom  card  <->  E. r 
r  We  A )
Distinct variable group:    A, r

Proof of Theorem ween
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac8b 8201 . 2  |-  ( A  e.  dom  card  ->  E. r  r  We  A
)
2 weso 4711 . . . . 5  |-  ( r  We  A  ->  r  Or  A )
3 vex 2975 . . . . 5  |-  r  e. 
_V
4 soex 6521 . . . . 5  |-  ( ( r  Or  A  /\  r  e.  _V )  ->  A  e.  _V )
52, 3, 4sylancl 662 . . . 4  |-  ( r  We  A  ->  A  e.  _V )
65exlimiv 1688 . . 3  |-  ( E. r  r  We  A  ->  A  e.  _V )
7 unipw 4542 . . . . . 6  |-  U. ~P A  =  A
8 weeq2 4709 . . . . . 6  |-  ( U. ~P A  =  A  ->  ( r  We  U. ~P A  <->  r  We  A
) )
97, 8ax-mp 5 . . . . 5  |-  ( r  We  U. ~P A  <->  r  We  A )
109exbii 1634 . . . 4  |-  ( E. r  r  We  U. ~P A  <->  E. r  r  We  A )
1110biimpri 206 . . 3  |-  ( E. r  r  We  A  ->  E. r  r  We 
U. ~P A )
12 pwexg 4476 . . . . 5  |-  ( A  e.  _V  ->  ~P A  e.  _V )
13 dfac8c 8203 . . . . 5  |-  ( ~P A  e.  _V  ->  ( E. r  r  We 
U. ~P A  ->  E. f A. x  e. 
~P  A ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
1412, 13syl 16 . . . 4  |-  ( A  e.  _V  ->  ( E. r  r  We  U. ~P A  ->  E. f A. x  e.  ~P  A ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
15 dfac8a 8200 . . . 4  |-  ( A  e.  _V  ->  ( E. f A. x  e. 
~P  A ( x  =/=  (/)  ->  ( f `  x )  e.  x
)  ->  A  e.  dom  card ) )
1614, 15syld 44 . . 3  |-  ( A  e.  _V  ->  ( E. r  r  We  U. ~P A  ->  A  e.  dom  card ) )
176, 11, 16sylc 60 . 2  |-  ( E. r  r  We  A  ->  A  e.  dom  card )
181, 17impbii 188 1  |-  ( A  e.  dom  card  <->  E. r 
r  We  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2606   A.wral 2715   _Vcvv 2972   (/)c0 3637   ~Pcpw 3860   U.cuni 4091    Or wor 4640    We wwe 4678   dom cdm 4840   ` cfv 5418   cardccrd 8105
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-recs 6832  df-en 7311  df-card 8109
This theorem is referenced by:  ondomen  8207  dfac10  8306  zorn2lem7  8671  fpwwe  8813  canthnumlem  8815  canthp1lem2  8820  pwfseqlem4a  8828  pwfseqlem4  8829  fin2so  28416
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