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Theorem relelec 6784
Description: Membership in an equivalence class when  R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
Assertion
Ref Expression
relelec  |-  ( Rel 
R  ->  ( A  e.  [ B ] R  <->  B R A ) )

Proof of Theorem relelec
StepHypRef Expression
1 elex 2872 . . . 4  |-  ( A  e.  [ B ] R  ->  A  e.  _V )
2 ecexr 6749 . . . 4  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
31, 2jca 518 . . 3  |-  ( A  e.  [ B ] R  ->  ( A  e. 
_V  /\  B  e.  _V ) )
43adantl 452 . 2  |-  ( ( Rel  R  /\  A  e.  [ B ] R
)  ->  ( A  e.  _V  /\  B  e. 
_V ) )
5 brrelex12 4805 . . 3  |-  ( ( Rel  R  /\  B R A )  ->  ( B  e.  _V  /\  A  e.  _V ) )
65ancomd 438 . 2  |-  ( ( Rel  R  /\  B R A )  ->  ( A  e.  _V  /\  B  e.  _V ) )
7 elecg 6782 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  e.  [ B ] R  <->  B R A ) )
84, 6, 7pm5.21nd 868 1  |-  ( Rel 
R  ->  ( A  e.  [ B ] R  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1710   _Vcvv 2864   class class class wbr 4102   Rel wrel 4773   [cec 6742
This theorem is referenced by:  eqgid  14762  tgptsmscls  17928  topfneec  25615
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-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-cnv 4776  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-ec 6746
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