Theorem brun 4166

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 3364	. 2 |- ( <. A , B >. e. ( R u. S ) <-> ( <. A , B >. e. R \/ <. A , B >. e. S ) )
2		df-br 4115	. 2 |- ( A ( R u. S ) B <-> <. A , B >. e. ( R u. S ) )
3		df-br 4115	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4115	. . 3 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	orbi12i 772	. 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 716   e. wcel 2205   u. cun 3212   <.cop 3697   class class class wbr 4114

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-br 4115

This theorem is referenced by:  dmun  4968  qfto  5157  poleloe  5167  cnvun  5173  coundi  5269  coundir  5270  fununmo  5403  brdifun  6807  ltxrlt  8355  ltxr  10127