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Theorem undi 3213
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 3154 . . . 4  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
21orbi2i 689 . . 3  |-  ( ( x  e.  A  \/  x  e.  ( B  i^i  C ) )  <->  ( x  e.  A  \/  (
x  e.  B  /\  x  e.  C )
) )
3 ordi 740 . . 3  |-  ( ( x  e.  A  \/  ( x  e.  B  /\  x  e.  C
) )  <->  ( (
x  e.  A  \/  x  e.  B )  /\  ( x  e.  A  \/  x  e.  C
) ) )
4 elin 3154 . . . 4  |-  ( x  e.  ( ( A  u.  B )  i^i  ( A  u.  C
) )  <->  ( x  e.  ( A  u.  B
)  /\  x  e.  ( A  u.  C
) ) )
5 elun 3112 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
6 elun 3112 . . . . 5  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
75, 6anbi12i 441 . . . 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 178 . . 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 199 . 2  |-  ( ( x  e.  A  \/  x  e.  ( B  i^i  C ) )  <->  x  e.  ( ( A  u.  B )  i^i  ( A  u.  C )
) )
109uneqri 3113 1  |-  ( A  u.  ( B  i^i  C ) )  =  ( ( A  u.  B
)  i^i  ( A  u.  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    \/ wo 639    = wceq 1259    e. wcel 1409    u. cun 2943    i^i cin 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952
This theorem is referenced by:  undir  3215
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