Theorem brrelex12 4636

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

Step	Hyp	Ref	Expression
1		df-rel 4605	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 119	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	ssbrd 4019	. . 3 |- ( Rel R -> ( A R B -> A ( _V X. _V ) B ) )
4	3	imp 123	. 2 |- ( ( Rel R /\ A R B ) -> A ( _V X. _V ) B )
5		brxp 4629	. 2 |- ( A ( _V X. _V ) B <-> ( A e. _V /\ B e. _V ) )
6	4, 5	sylib 121	1 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2135   _Vcvv 2721   C_ wss 3111   class class class wbr 3976   X. cxp 4596   Rel wrel 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-xp 4604  df-rel 4605

This theorem is referenced by:  brrelex1  4637  brrelex  4638  brrelex2  4639  brrelex12i  4640  relbrcnvg  4977  ovprc  5868  ersym  6504  relelec  6532  encv  6703