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Theorem elecg 6631
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 4992 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( A  e.  ( R " { B } )  <->  <. B ,  A >.  e.  R ) )
21ancoms 441 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  ( R " { B } )  <->  <. B ,  A >.  e.  R ) )
3 df-ec 6595 . . 3  |-  [ B ] R  =  ( R " { B }
)
43eleq2i 2320 . 2  |-  ( A  e.  [ B ] R 
<->  A  e.  ( R
" { B }
) )
5 df-br 3964 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
62, 4, 53bitr4g 281 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  [ B ] R  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621   {csn 3581   <.cop 3584   class class class wbr 3963   "cima 4629   [cec 6591
This theorem is referenced by:  elec  6632  relelec  6633  ecdmn0  6635  erth  6637  erdisj  6640  qsel  6671  orbsta  14694  sylow2alem1  14855  sylow2blem1  14858  sylow3lem3  14867  efgi2  14961  tgpconcompeqg  17721  xmetec  17907  blpnfctr  17909  xmetresbl  17910  xrsblre  18244  pdiveql  25500
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-xp 4640  df-cnv 4642  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-ec 6595
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