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Theorem brun 3983
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 3218 . 2  |-  ( <. A ,  B >.  e.  ( R  u.  S
)  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
2 df-br 3934 . 2  |-  ( A ( R  u.  S
) B  <->  <. A ,  B >.  e.  ( R  u.  S ) )
3 df-br 3934 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 3934 . . 3  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4orbi12i 754 . 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 698    e. wcel 1481    u. cun 3070   <.cop 3531   class class class wbr 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-un 3076  df-br 3934
This theorem is referenced by:  dmun  4750  qfto  4932  poleloe  4942  cnvun  4948  coundi  5044  coundir  5045  brdifun  6460  ltxrlt  7850  ltxr  9588
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