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Theorem brun 4066
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 3288 . 2  |-  ( <. A ,  B >.  e.  ( R  u.  S
)  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
2 df-br 4016 . 2  |-  ( A ( R  u.  S
) B  <->  <. A ,  B >.  e.  ( R  u.  S ) )
3 df-br 4016 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4016 . . 3  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4orbi12i 765 . 2  |-  ( ( A R B  \/  A S B )  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
61, 2, 53bitr4i 212 1  |-  ( A ( R  u.  S
) B  <->  ( A R B  \/  A S B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    \/ wo 709    e. wcel 2158    u. cun 3139   <.cop 3607   class class class wbr 4015
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-br 4016
This theorem is referenced by:  dmun  4846  qfto  5030  poleloe  5040  cnvun  5046  coundi  5142  coundir  5143  brdifun  6576  ltxrlt  8037  ltxr  9789
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