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

Theorem brrelex 4919
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 4918 . 2  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
21simpld 447 1  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   _Vcvv 2958   class class class wbr 4215   Rel wrel 4886
This theorem is referenced by:  brrelexi  4921  posn  4949  frsn  4951  releldm  5105  relelrn  5106  relimasn  5230  funmo  5473  ertr  6923  spwpr4  14668  dirtr  14686  vdgrun  21677  vdgrfiun  21678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888
  Copyright terms: Public domain W3C validator