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Theorem brrelex12 5029
Description: A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brrelex12  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )

Proof of Theorem brrelex12
StepHypRef Expression
1 df-rel 4999 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 194 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32ssbrd 4481 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  A ( _V  X.  _V ) B ) )
43imp 429 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A
( _V  X.  _V ) B )
5 brxp 5022 . 2  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
64, 5sylib 196 1  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   _Vcvv 3106    C_ wss 3469   class class class wbr 4440    X. cxp 4990   Rel wrel 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999
This theorem is referenced by:  brrelex  5030  brrelex2  5031  relbrcnvg  5366  ovprc  6302  oprabv  6320  brovex  6940  ersym  7313  relelec  7342  encv  7514  bren  7515  fsuppunbi  7839  fpwwe2lem2  8999  fpwwelem  9012  fpwwe  9013  isstruct2  14488  brssc  15033  cofuval2  15103  isfull  15126  isfth  15130  isnat  15163  pslem  15682  efgrelexlema  16556  frgpuplem  16579  dvdsr  17072  tpsexOLD  19180  ulmval  22502  perpln1  23788  perpln2  23789  iseupa  24627  rngoablo2  25086  opelco3  28771  aovprc  31695  aovrcl  31696
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