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Theorem ecdmn0 6950
Description: A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecdmn0  |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )

Proof of Theorem ecdmn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2966 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 n0 3639 . . 3  |-  ( [ A ] R  =/=  (/) 
<->  E. x  x  e. 
[ A ] R
)
3 ecexr 6913 . . . 4  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
43exlimiv 1645 . . 3  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
52, 4sylbi 189 . 2  |-  ( [ A ] R  =/=  (/)  ->  A  e.  _V )
6 vex 2961 . . . . 5  |-  x  e. 
_V
7 elecg 6946 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
86, 7mpan 653 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  [ A ] R  <->  A R x ) )
98exbidv 1637 . . 3  |-  ( A  e.  _V  ->  ( E. x  x  e.  [ A ] R  <->  E. x  A R x ) )
102a1i 11 . . 3  |-  ( A  e.  _V  ->  ( [ A ] R  =/=  (/) 
<->  E. x  x  e. 
[ A ] R
) )
11 eldmg 5068 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
129, 10, 113bitr4rd 279 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) ) )
131, 5, 12pm5.21nii 344 1  |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    e. wcel 1726    =/= wne 2601   _Vcvv 2958   (/)c0 3630   class class class wbr 4215   dom cdm 4881   [cec 6906
This theorem is referenced by:  ereldm  6951  elqsn0  6976  ecelqsdm  6977  eceqoveq  7012  divsfval  13777  sylow1lem5  15241  vitalilem2  19506  vitalilem3  19507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-ec 6910
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