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Theorem elecg 6698
Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
elecg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  [ B ] R  <->  B R A ) )

Proof of Theorem elecg
StepHypRef Expression
1 elimasng 5039 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( A  e.  ( R " { B } )  <->  <. B ,  A >.  e.  R ) )
21ancoms 439 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  ( R " { B } )  <->  <. B ,  A >.  e.  R ) )
3 df-ec 6662 . . 3  |-  [ B ] R  =  ( R " { B }
)
43eleq2i 2347 . 2  |-  ( A  e.  [ B ] R 
<->  A  e.  ( R
" { B }
) )
5 df-br 4024 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
62, 4, 53bitr4g 279 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  [ B ] R  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   {csn 3640   <.cop 3643   class class class wbr 4023   "cima 4692   [cec 6658
This theorem is referenced by:  elec  6699  relelec  6700  ecdmn0  6702  erth  6704  erdisj  6707  qsel  6738  orbsta  14767  sylow2alem1  14928  sylow2blem1  14931  sylow3lem3  14940  efgi2  15034  tgpconcompeqg  17794  xmetec  17980  blpnfctr  17982  xmetresbl  17983  xrsblre  18317  pdiveql  26168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662
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