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Theorem dfac12a 7954
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 3304 . . . 4  |-  dom  card  C_ 
_V
2 eqss 3299 . . . 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 7943 . . 3  |-  (CHOICE  <->  dom  card  =  _V )
5 unir1 7665 . . . 4  |-  U. ( R1 " On )  =  _V
65sseq1i 3308 . . 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 7952 . 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 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892    C_ wss 3256   ~Pcpw 3735   U.cuni 3950   Oncon0 4515   dom cdm 4811   "cima 4814   R1cr1 7614   cardccrd 7748  CHOICEwac 7922
This theorem is referenced by:  dfac12  7955
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-reg 7486  ax-inf2 7522
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-recs 6562  df-rdg 6597  df-oadd 6657  df-omul 6658  df-er 6834  df-en 7039  df-dom 7040  df-oi 7405  df-har 7452  df-r1 7616  df-rank 7617  df-card 7752  df-ac 7923
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