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Theorem ecdmn0 6844
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 2881 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 n0 3552 . . 3  |-  ( [ A ] R  =/=  (/) 
<->  E. x  x  e. 
[ A ] R
)
3 ecexr 6807 . . . 4  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
43exlimiv 1639 . . 3  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
52, 4sylbi 187 . 2  |-  ( [ A ] R  =/=  (/)  ->  A  e.  _V )
6 vex 2876 . . . . 5  |-  x  e. 
_V
7 elecg 6840 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
86, 7mpan 651 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  [ A ] R  <->  A R x ) )
98exbidv 1631 . . 3  |-  ( A  e.  _V  ->  ( E. x  x  e.  [ A ] R  <->  E. x  A R x ) )
102a1i 10 . . 3  |-  ( A  e.  _V  ->  ( [ A ] R  =/=  (/) 
<->  E. x  x  e. 
[ A ] R
) )
11 eldmg 4977 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
129, 10, 113bitr4rd 277 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) ) )
131, 5, 12pm5.21nii 342 1  |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1546    e. wcel 1715    =/= wne 2529   _Vcvv 2873   (/)c0 3543   class class class wbr 4125   dom cdm 4792   [cec 6800
This theorem is referenced by:  ereldm  6845  elqsn0  6870  ecelqsdm  6871  eceqoveq  6906  divsfval  13659  sylow1lem5  15123  vitalilem2  19179  vitalilem3  19180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-xp 4798  df-cnv 4800  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-ec 6804
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