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Theorem brrelex2 4890
Description: A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brrelex2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )

Proof of Theorem brrelex2
StepHypRef Expression
1 brrelex12 4888 . 2  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
21simprd 464 1  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1868   _Vcvv 3081   class class class wbr 4420   Rel wrel 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-xp 4856  df-rel 4857
This theorem is referenced by:  brrelex2i  4892  releldm  5083  relelrn  5084  elrelimasn  5208  funbrfv  5916  relbrtpos  6989  ertr  7383  erth  7413  pslem  16440  opeldifid  28200  frege124d  36213  frege133d  36217
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