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Theorem brun 4085
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
StepHypRef Expression
1 elun 3329 . 2  |-  ( <. A ,  B >.  e.  ( R  u.  S
)  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
2 df-br 4040 . 2  |-  ( A ( R  u.  S
) B  <->  <. A ,  B >.  e.  ( R  u.  S ) )
3 df-br 4040 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4040 . . 3  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4orbi12i 507 . 2  |-  ( ( A R B  \/  A S B )  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
61, 2, 53bitr4i 268 1  |-  ( A ( R  u.  S
) B  <->  ( A R B  \/  A S B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    e. wcel 1696    u. cun 3163   <.cop 3656   class class class wbr 4039
This theorem is referenced by:  dmun  4901  qfto  5080  poleloe  5093  cnvun  5102  coundi  5190  coundir  5191  brdifun  6703  fpwwe2lem13  8280  ltxrlt  8909  ltxr  10473  dfle2  10497  dfso2  24182  dfon3  24503  brcup  24549  dfrdg4  24560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-br 4040
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