Theorem brun 3921

Description: The union of two binary relations. (Contributed by NM, 21-Dec-2008.)

Assertion

Ref	Expression
brun	|- ( A ( R u. S ) B <-> ( A R B \/ A S B ) )

Proof of Theorem brun

Step	Hyp	Ref	Expression
1		elun 3164	. 2 |- ( <. A , B >. e. ( R u. S ) <-> ( <. A , B >. e. R \/ <. A , B >. e. S ) )
2		df-br 3876	. 2 |- ( A ( R u. S ) B <-> <. A , B >. e. ( R u. S ) )
3		df-br 3876	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 3876	. . 3 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	orbi12i 722	. 2 |- ( ( A R B \/ A S B ) <-> ( <. A , B >. e. R \/ <. A , B >. e. S ) )
6	1, 2, 5	3bitr4i 211	1 |- ( A ( R u. S ) B <-> ( A R B \/ A S B ) )

Syntax hints:   <-> wb 104   \/ wo 670   e. wcel 1448   u. cun 3019   <.cop 3477   class class class wbr 3875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-br 3876

This theorem is referenced by:  dmun  4684  qfto  4864  poleloe  4874  cnvun  4880  coundi  4976  coundir  4977  brdifun  6386  ltxrlt  7702  ltxr  9403