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Theorem elecg 6714
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 5055 . . 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 6678 . . 3  |-  [ B ] R  =  ( R " { B }
)
43eleq2i 2360 . 2  |-  ( A  e.  [ B ] R 
<->  A  e.  ( R
" { B }
) )
5 df-br 4040 . 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 1696   {csn 3653   <.cop 3656   class class class wbr 4039   "cima 4708   [cec 6674
This theorem is referenced by:  elec  6715  relelec  6716  ecdmn0  6718  erth  6720  erdisj  6723  qsel  6754  orbsta  14783  sylow2alem1  14944  sylow2blem1  14947  sylow3lem3  14956  efgi2  15050  tgpconcompeqg  17810  xmetec  17996  blpnfctr  17998  xmetresbl  17999  xrsblre  18333  pdiveql  26271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-ec 6678
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