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

Theorem brrelex 4743
Description: A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brrelex  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )

Proof of Theorem brrelex
StepHypRef Expression
1 brrelex12 4742 . 2  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
21simpld 445 1  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   Rel wrel 4710
This theorem is referenced by:  brrelexi  4745  posn  4774  frsn  4776  releldm  4927  relelrn  4928  relimasn  5052  funmo  5287  ertr  6691  spwpr4  14356  dirtr  14374  vdgrun  23908  preotr2  25338  nfwpr4c  25388
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-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-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-rel 4712
  Copyright terms: Public domain W3C validator