Theorem brun 4094

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 3313	. 2 |- ( <. A , B >. e. ( R u. S ) <-> ( <. A , B >. e. R \/ <. A , B >. e. S ) )
2		df-br 4044	. 2 |- ( A ( R u. S ) B <-> <. A , B >. e. ( R u. S ) )
3		df-br 4044	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4044	. . 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 2175   u. cun 3163   <.cop 3635   class class class wbr 4043

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-br 4044

This theorem is referenced by:  dmun  4884  qfto  5071  poleloe  5081  cnvun  5087  coundi  5183  coundir  5184  fununmo  5315  brdifun  6646  ltxrlt  8137  ltxr  9896