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Theorem undi 3745
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 3687 . . . 4  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
21orbi2i 519 . . 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 3687 . . . 4  |-  ( x  e.  ( ( A  u.  B )  i^i  ( A  u.  C
) )  <->  ( x  e.  ( A  u.  B
)  /\  x  e.  ( A  u.  C
) ) )
5 elun 3645 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
6 elun 3645 . . . . 5  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
75, 6anbi12i 697 . . . 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 3646 1  |-  ( A  u.  ( B  i^i  C ) )  =  ( ( A  u.  B
)  i^i  ( A  u.  C ) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3474    i^i cin 3475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-in 3483
This theorem is referenced by:  undir  3747  dfif4  3954  dfif5  3955
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