Theorem brrelex12 4642

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 4611	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 119	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	ssbrd 4025	. . 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 4635	. 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 2136   _Vcvv 2726   C_ wss 3116   class class class wbr 3982   X. cxp 4602   Rel wrel 4609

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611

This theorem is referenced by:  brrelex1  4643  brrelex  4644  brrelex2  4645  brrelex12i  4646  relbrcnvg  4983  ovprc  5877  ersym  6513  relelec  6541  encv  6712