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Theorem undi 3580
Description: Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undi  |-  ( A  u.  ( B  i^i  C ) )  =  ( ( A  u.  B
)  i^i  ( A  u.  C ) )

Proof of Theorem undi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3522 . . . 4  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
21orbi2i 506 . . 3  |-  ( ( x  e.  A  \/  x  e.  ( B  i^i  C ) )  <->  ( x  e.  A  \/  (
x  e.  B  /\  x  e.  C )
) )
3 ordi 835 . . 3  |-  ( ( x  e.  A  \/  ( x  e.  B  /\  x  e.  C
) )  <->  ( (
x  e.  A  \/  x  e.  B )  /\  ( x  e.  A  \/  x  e.  C
) ) )
4 elin 3522 . . . 4  |-  ( x  e.  ( ( A  u.  B )  i^i  ( A  u.  C
) )  <->  ( x  e.  ( A  u.  B
)  /\  x  e.  ( A  u.  C
) ) )
5 elun 3480 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
6 elun 3480 . . . . 5  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
75, 6anbi12i 679 . . . 4  |-  ( ( x  e.  ( A  u.  B )  /\  x  e.  ( A  u.  C ) )  <->  ( (
x  e.  A  \/  x  e.  B )  /\  ( x  e.  A  \/  x  e.  C
) ) )
84, 7bitr2i 242 . . 3  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ( x  e.  A  \/  x  e.  C ) )  <->  x  e.  ( ( A  u.  B )  i^i  ( A  u.  C )
) )
92, 3, 83bitri 263 . 2  |-  ( ( x  e.  A  \/  x  e.  ( B  i^i  C ) )  <->  x  e.  ( ( A  u.  B )  i^i  ( A  u.  C )
) )
109uneqri 3481 1  |-  ( A  u.  ( B  i^i  C ) )  =  ( ( A  u.  B
)  i^i  ( A  u.  C ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3310    i^i cin 3311
This theorem is referenced by:  undir  3582  dfif4  3742  dfif5  3743
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-in 3319
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