MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elec Unicode version

Theorem elec 6903
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 6902 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  e.  [ B ] R  <->  B R A ) )
41, 2, 3mp2an 654 1  |-  ( A  e.  [ B ] R 
<->  B R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1721   _Vcvv 2916   class class class wbr 4172   [cec 6862
This theorem is referenced by:  ecid  6928  sylow2alem2  15207  sylow2a  15208  sylow2blem1  15209  efgval2  15311  efgrelexlemb  15337  efgcpbllemb  15342  frgpnabllem1  15439  tgpconcomp  18095  divstgphaus  18105  vitalilem2  19454  vitalilem3  19455  isbndx  26381  prtlem10  26604  prtlem19  26617  prter3  26621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866
  Copyright terms: Public domain W3C validator