Theorem brrelex 5156

Description: A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brrelex	|- ( ( Rel R /\ A R B ) -> A e. _V )

Proof of Theorem brrelex

Step	Hyp	Ref	Expression
1		brrelex12 5155	. 2 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
2	1	simpld 475	1 |- ( ( Rel R /\ A R B ) -> A e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  brrelexi  5158  posn  5187  frsn  5189  releldm  5358  relelrn  5359  relimasn  5488  funmo  5904  ertr  7757  dirtr  17236  frege129d  38055  nnfoctb  39213  clim2d  39905  climfv  39923  meadjiun  40683  caragenunicl  40738