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Theorem ecdmn0 6906
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 2924 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 n0 3597 . . 3  |-  ( [ A ] R  =/=  (/) 
<->  E. x  x  e. 
[ A ] R
)
3 ecexr 6869 . . . 4  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
43exlimiv 1641 . . 3  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
52, 4sylbi 188 . 2  |-  ( [ A ] R  =/=  (/)  ->  A  e.  _V )
6 vex 2919 . . . . 5  |-  x  e. 
_V
7 elecg 6902 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
86, 7mpan 652 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  [ A ] R  <->  A R x ) )
98exbidv 1633 . . 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 5024 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
129, 10, 113bitr4rd 278 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) ) )
131, 5, 12pm5.21nii 343 1  |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1547    e. wcel 1721    =/= wne 2567   _Vcvv 2916   (/)c0 3588   class class class wbr 4172   dom cdm 4837   [cec 6862
This theorem is referenced by:  ereldm  6907  elqsn0  6932  ecelqsdm  6933  eceqoveq  6968  divsfval  13727  sylow1lem5  15191  vitalilem2  19454  vitalilem3  19455
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866
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