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Theorem brrelex12 4661
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 4630 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 120 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32ssbrd 4043 . . 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 4654 . 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 2148   _Vcvv 2737    C_ wss 3129   class class class wbr 4000    X. cxp 4621   Rel wrel 4628
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630
This theorem is referenced by:  brrelex1  4662  brrelex  4663  brrelex2  4664  brrelex12i  4665  relbrcnvg  5003  ovprc  5904  ersym  6541  relelec  6569  encv  6740  dvdsrd  13085
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