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Theorem brun 4199
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 3431 . 2  |-  ( <. A ,  B >.  e.  ( R  u.  S
)  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
2 df-br 4154 . 2  |-  ( A ( R  u.  S
) B  <->  <. A ,  B >.  e.  ( R  u.  S ) )
3 df-br 4154 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4154 . . 3  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4orbi12i 508 . 2  |-  ( ( A R B  \/  A S B )  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
61, 2, 53bitr4i 269 1  |-  ( A ( R  u.  S
) B  <->  ( A R B  \/  A S B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    e. wcel 1717    u. cun 3261   <.cop 3760   class class class wbr 4153
This theorem is referenced by:  dmun  5016  qfto  5195  poleloe  5208  cnvun  5217  coundi  5311  coundir  5312  brdifun  6868  fpwwe2lem13  8450  ltxrlt  9079  ltxr  10647  dfle2  10672  dfso2  25135  dfon3  25456  brcup  25502  dfrdg4  25513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-un 3268  df-br 4154
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