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Theorem elec 6715
Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
Hypotheses
Ref Expression
elec.1  |-  A  e. 
_V
elec.2  |-  B  e. 
_V
Assertion
Ref Expression
elec  |-  ( A  e.  [ B ] R 
<->  B R A )

Proof of Theorem elec
StepHypRef Expression
1 elec.1 . 2  |-  A  e. 
_V
2 elec.2 . 2  |-  B  e. 
_V
3 elecg 6714 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  e.  [ B ] R  <->  B R A ) )
41, 2, 3mp2an 653 1  |-  ( A  e.  [ B ] R 
<->  B R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   [cec 6674
This theorem is referenced by:  ecid  6740  sylow2alem2  14945  sylow2a  14946  sylow2blem1  14947  efgval2  15049  efgrelexlemb  15075  efgcpbllemb  15080  frgpnabllem1  15177  tgpconcomp  17811  divstgphaus  17821  vitalilem2  18980  vitalilem3  18981  isbndx  26609  prtlem10  26836  prtlem19  26849  prter3  26853
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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