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Theorem brun 3921
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 3164 . 2  |-  ( <. A ,  B >.  e.  ( R  u.  S
)  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
2 df-br 3876 . 2  |-  ( A ( R  u.  S
) B  <->  <. A ,  B >.  e.  ( R  u.  S ) )
3 df-br 3876 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 3876 . . 3  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4orbi12i 722 . 2  |-  ( ( A R B  \/  A S B )  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
61, 2, 53bitr4i 211 1  |-  ( A ( R  u.  S
) B  <->  ( A R B  \/  A S B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 670    e. wcel 1448    u. cun 3019   <.cop 3477   class class class wbr 3875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-br 3876
This theorem is referenced by:  dmun  4684  qfto  4864  poleloe  4874  cnvun  4880  coundi  4976  coundir  4977  brdifun  6386  ltxrlt  7702  ltxr  9403
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