Theorem brun

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

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

Proof

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