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Theorem brun 4250
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 3480 . 2  |-  ( <. A ,  B >.  e.  ( R  u.  S
)  <->  ( <. A ,  B >.  e.  R  \/  <. A ,  B >.  e.  S ) )
2 df-br 4205 . 2  |-  ( A ( R  u.  S
) B  <->  <. A ,  B >.  e.  ( R  u.  S ) )
3 df-br 4205 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4205 . . 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 1725    u. cun 3310   <.cop 3809   class class class wbr 4204
This theorem is referenced by:  dmun  5068  qfto  5247  poleloe  5260  cnvun  5269  coundi  5363  coundir  5364  brdifun  6924  fpwwe2lem13  8509  ltxrlt  9138  ltxr  10707  dfle2  10732  dfso2  25369  dfon3  25729  brcup  25776  dfrdg4  25787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-br 4205
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