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Theorem dfac10b 8519
Description: Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 8497). (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
dfac10b  |-  (CHOICE  <->  (  ~~  " On )  =  _V )

Proof of Theorem dfac10b
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3116 . . . . 5  |-  x  e. 
_V
21elima 5342 . . . 4  |-  ( x  e.  (  ~~  " On ) 
<->  E. y  e.  On  y  ~~  x )
32bicomi 202 . . 3  |-  ( E. y  e.  On  y  ~~  x  <->  x  e.  (  ~~  " On ) )
43albii 1620 . 2  |-  ( A. x E. y  e.  On  y  ~~  x  <->  A. x  x  e.  (  ~~  " On ) )
5 dfac10c 8518 . 2  |-  (CHOICE  <->  A. x E. y  e.  On  y  ~~  x )
6 eqv 3801 . 2  |-  ( ( 
~~  " On )  =  _V  <->  A. x  x  e.  (  ~~  " On ) )
74, 5, 63bitr4i 277 1  |-  (CHOICE  <->  (  ~~  " On )  =  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1377    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   class class class wbr 4447   Oncon0 4878   "cima 5002    ~~ cen 7513  CHOICEwac 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-recs 7042  df-en 7517  df-card 8320  df-ac 8497
This theorem is referenced by:  axac10  30607
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