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Theorem dfac12a 8021
Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
dfac12a  |-  (CHOICE  <->  A. x  e.  On  ~P x  e. 
dom  card )

Proof of Theorem dfac12a
StepHypRef Expression
1 ssv 3361 . . . 4  |-  dom  card  C_ 
_V
2 eqss 3356 . . . 4  |-  ( dom 
card  =  _V  <->  ( dom  card  C_  _V  /\  _V  C_  dom  card ) )
31, 2mpbiran 885 . . 3  |-  ( dom 
card  =  _V  <->  _V  C_  dom  card )
4 dfac10 8010 . . 3  |-  (CHOICE  <->  dom  card  =  _V )
5 unir1 7732 . . . 4  |-  U. ( R1 " On )  =  _V
65sseq1i 3365 . . 3  |-  ( U. ( R1 " On ) 
C_  dom  card  <->  _V  C_  dom  card )
73, 4, 63bitr4i 269 . 2  |-  (CHOICE  <->  U. ( R1 " On )  C_  dom  card )
8 dfac12r 8019 . 2  |-  ( A. x  e.  On  ~P x  e.  dom  card  <->  U. ( R1 " On )  C_  dom  card )
97, 8bitr4i 244 1  |-  (CHOICE  <->  A. x  e.  On  ~P x  e. 
dom  card )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   A.wral 2698   _Vcvv 2949    C_ wss 3313   ~Pcpw 3792   U.cuni 4008   Oncon0 4574   dom cdm 4871   "cima 4874   R1cr1 7681   cardccrd 7815  CHOICEwac 7989
This theorem is referenced by:  dfac12  8022
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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-reg 7553  ax-inf2 7589
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-riota 6542  df-recs 6626  df-rdg 6661  df-oadd 6721  df-omul 6722  df-er 6898  df-en 7103  df-dom 7104  df-oi 7472  df-har 7519  df-r1 7683  df-rank 7684  df-card 7819  df-ac 7990
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