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Theorem dfac10c 7851
Description: Axiom of Choice equivalent: every set is equinumerous to an ordinal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
dfac10c  |-  (CHOICE  <->  A. x E. y  e.  On  y  ~~  x )
Distinct variable group:    x, y

Proof of Theorem dfac10c
StepHypRef Expression
1 dfac10 7850 . 2  |-  (CHOICE  <->  dom  card  =  _V )
2 eqv 3546 . 2  |-  ( dom 
card  =  _V  <->  A. x  x  e.  dom  card )
3 isnum2 7665 . . 3  |-  ( x  e.  dom  card  <->  E. y  e.  On  y  ~~  x
)
43albii 1566 . 2  |-  ( A. x  x  e.  dom  card  <->  A. x E. y  e.  On  y  ~~  x
)
51, 2, 43bitri 262 1  |-  (CHOICE  <->  A. x E. y  e.  On  y  ~~  x )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1540    = wceq 1642    e. wcel 1710   E.wrex 2620   _Vcvv 2864   class class class wbr 4102   Oncon0 4471   dom cdm 4768    ~~ cen 6945   cardccrd 7655  CHOICEwac 7829
This theorem is referenced by:  dfac10b  7852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-suc 4477  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-riota 6388  df-recs 6472  df-en 6949  df-card 7659  df-ac 7830
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