Theorem brrelex12 4714

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 4683	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 120	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	ssbrd 4088	. . 3 |- ( Rel R -> ( A R B -> A ( _V X. _V ) B ) )
4	3	imp 124	. 2 |- ( ( Rel R /\ A R B ) -> A ( _V X. _V ) B )
5		brxp 4707	. 2 |- ( A ( _V X. _V ) B <-> ( A e. _V /\ B e. _V ) )
6	4, 5	sylib 122	1 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   _Vcvv 2772   C_ wss 3166   class class class wbr 4045   X. cxp 4674   Rel wrel 4681

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683

This theorem is referenced by:  brrelex1  4715  brrelex  4716  brrelex2  4717  brrelex12i  4718  relbrcnvg  5062  ovprc  5982  ersym  6634  relelec  6664  encv  6835  dvdsrd  13889