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Theorem brrelex12 4697
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 4666 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 120 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32ssbrd 4072 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  A ( _V  X.  _V ) B ) )
43imp 124 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A
( _V  X.  _V ) B )
5 brxp 4690 . 2  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
64, 5sylib 122 1  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2164   _Vcvv 2760    C_ wss 3153   class class class wbr 4029    X. cxp 4657   Rel wrel 4664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666
This theorem is referenced by:  brrelex1  4698  brrelex  4699  brrelex2  4700  brrelex12i  4701  relbrcnvg  5044  ovprc  5953  ersym  6599  relelec  6629  encv  6800  dvdsrd  13590
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