Theorem brun 4066

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 3288	. 2 |- ( <. A , B >. e. ( R u. S ) <-> ( <. A , B >. e. R \/ <. A , B >. e. S ) )
2		df-br 4016	. 2 |- ( A ( R u. S ) B <-> <. A , B >. e. ( R u. S ) )
3		df-br 4016	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4016	. . 3 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	orbi12i 765	. 2 |- ( ( A R B \/ A S B ) <-> ( <. A , B >. e. R \/ <. A , B >. e. S ) )
6	1, 2, 5	3bitr4i 212	1 |- ( A ( R u. S ) B <-> ( A R B \/ A S B ) )

Syntax hints:   <-> wb 105   \/ wo 709   e. wcel 2158   u. cun 3139   <.cop 3607   class class class wbr 4015

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-br 4016

This theorem is referenced by:  dmun  4846  qfto  5030  poleloe  5040  cnvun  5046  coundi  5142  coundir  5143  brdifun  6576  ltxrlt  8037  ltxr  9789