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

Theorem elec 6653
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 6652 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  e.  [ B ] R  <->  B R A ) )
41, 2, 3mp2an 656 1  |-  ( A  e.  [ B ] R 
<->  B R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1621   _Vcvv 2757   class class class wbr 3983   [cec 6612
This theorem is referenced by:  ecid  6678  sylow2alem2  14877  sylow2a  14878  sylow2blem1  14879  efgval2  14981  efgrelexlemb  15007  efgcpbllemb  15012  frgpnabllem1  15109  tgpconcomp  17743  divstgphaus  17753  vitalilem2  18912  vitalilem3  18913  isbndx  25859  prtlem10  26086  prtlem19  26099  prter3  26103
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 4101  ax-nul 4109  ax-pr 4172
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 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-opab 4038  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-ec 6616
  Copyright terms: Public domain W3C validator