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Theorem undi 3670
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 3601 . . . 4  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
21orbi2i 517 . . 3  |-  ( ( x  e.  A  \/  x  e.  ( B  i^i  C ) )  <->  ( x  e.  A  \/  (
x  e.  B  /\  x  e.  C )
) )
3 ordi 862 . . 3  |-  ( ( x  e.  A  \/  ( x  e.  B  /\  x  e.  C
) )  <->  ( (
x  e.  A  \/  x  e.  B )  /\  ( x  e.  A  \/  x  e.  C
) ) )
4 elin 3601 . . . 4  |-  ( x  e.  ( ( A  u.  B )  i^i  ( A  u.  C
) )  <->  ( x  e.  ( A  u.  B
)  /\  x  e.  ( A  u.  C
) ) )
5 elun 3559 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
6 elun 3559 . . . . 5  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
75, 6anbi12i 695 . . . 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 250 . . 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 271 . 2  |-  ( ( x  e.  A  \/  x  e.  ( B  i^i  C ) )  <->  x  e.  ( ( A  u.  B )  i^i  ( A  u.  C )
) )
109uneqri 3560 1  |-  ( A  u.  ( B  i^i  C ) )  =  ( ( A  u.  B
)  i^i  ( A  u.  C ) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1826    u. cun 3387    i^i cin 3388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-un 3394  df-in 3396
This theorem is referenced by:  undir  3672  dfif4  3872  dfif5  3873
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