Theorem brun 4106

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 3318	. 2 |- ( <. A , B >. e. ( R u. S ) <-> ( <. A , B >. e. R \/ <. A , B >. e. S ) )
2		df-br 4055	. 2 |- ( A ( R u. S ) B <-> <. A , B >. e. ( R u. S ) )
3		df-br 4055	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4055	. . 3 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	orbi12i 766	. 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 710   e. wcel 2177   u. cun 3168   <.cop 3641   class class class wbr 4054

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-br 4055

This theorem is referenced by:  dmun  4899  qfto  5086  poleloe  5096  cnvun  5102  coundi  5198  coundir  5199  fununmo  5330  brdifun  6665  ltxrlt  8168  ltxr  9927