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Theorem brrelex12 4742
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 4712 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 186 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32ssbrd 4080 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  A ( _V  X.  _V ) B ) )
43imp 418 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A
( _V  X.  _V ) B )
5 brxp 4736 . 2  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
64, 5sylib 188 1  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    X. cxp 4703   Rel wrel 4710
This theorem is referenced by:  brrelex  4743  brrelex2  4744  relbrcnvg  5068  ovprc  5901  ersym  6688  relelec  6716  bren  6887  fpwwe2lem2  8270  fpwwelem  8283  fpwwe  8284  isstruct2  13173  brssc  13707  isfunc  13754  cofuval2  13777  isfull  13800  isfth  13804  isnat  13837  pslem  14331  efgrelexlema  15074  frgpuplem  15097  dvdsr  15444  tpsexOLD  16673  ulmval  19775  rngoablo2  21105  iseupa  23896  aovprc  28156  aovrcl  28157  brovex  28205
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
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