Theorem brrelex 4728

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 4726	. 2 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
2	1	simpld 112	1 |- ( ( Rel R /\ A R B ) -> A e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2177   _Vcvv 2773   class class class wbr 4054   Rel wrel 4693

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4055  df-opab 4117  df-xp 4694  df-rel 4695

This theorem is referenced by:  releldm  4927  relelrn  4928  funmo  5300  ertr  6653