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Theorem ennum 7834
Description: Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
ennum  |-  ( A 
~~  B  ->  ( A  e.  dom  card  <->  B  e.  dom  card ) )

Proof of Theorem ennum
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 enen2 7248 . . 3  |-  ( A 
~~  B  ->  (
x  ~~  A  <->  x  ~~  B ) )
21rexbidv 2726 . 2  |-  ( A 
~~  B  ->  ( E. x  e.  On  x  ~~  A  <->  E. x  e.  On  x  ~~  B
) )
3 isnum2 7832 . 2  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
4 isnum2 7832 . 2  |-  ( B  e.  dom  card  <->  E. x  e.  On  x  ~~  B
)
52, 3, 43bitr4g 280 1  |-  ( A 
~~  B  ->  ( A  e.  dom  card  <->  B  e.  dom  card ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725   E.wrex 2706   class class class wbr 4212   Oncon0 4581   dom cdm 4878    ~~ cen 7106   cardccrd 7822
This theorem is referenced by:  carden2b  7854  dfac12lem3  8025  dfac12k  8027  qnnen  12813  cygctb  15501
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-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-er 6905  df-en 7110  df-card 7826
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