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Theorem indi 3218
Description: Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
indi  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )

Proof of Theorem indi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 andi 765 . . . 4  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( (
x  e.  A  /\  x  e.  B )  \/  ( x  e.  A  /\  x  e.  C
) ) )
2 elin 3156 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3 elin 3156 . . . . 5  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
42, 3orbi12i 714 . . . 4  |-  ( ( x  e.  ( A  i^i  B )  \/  x  e.  ( A  i^i  C ) )  <-> 
( ( x  e.  A  /\  x  e.  B )  \/  (
x  e.  A  /\  x  e.  C )
) )
51, 4bitr4i 185 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  ( A  i^i  B
)  \/  x  e.  ( A  i^i  C
) ) )
6 elun 3114 . . . 4  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
76anbi2i 445 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  \/  x  e.  C )
) )
8 elun 3114 . . 3  |-  ( x  e.  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  <->  ( x  e.  ( A  i^i  B
)  \/  x  e.  ( A  i^i  C
) ) )
95, 7, 83bitr4i 210 . 2  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  x  e.  ( ( A  i^i  B )  u.  ( A  i^i  C ) ) )
109ineqri 3166 1  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 662    = wceq 1285    e. wcel 1434    u. cun 2972    i^i cin 2973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980
This theorem is referenced by:  indir  3220  undisj2  3309  disjssun  3314  difdifdirss  3334  disjpr2  3464  diftpsn3  3535  resundi  4653
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