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Theorem indi 3574
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 838 . . . 4  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( (
x  e.  A  /\  x  e.  B )  \/  ( x  e.  A  /\  x  e.  C
) ) )
2 elin 3517 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3 elin 3517 . . . . 5  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
42, 3orbi12i 508 . . . 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 244 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  ( A  i^i  B
)  \/  x  e.  ( A  i^i  C
) ) )
6 elun 3475 . . . 4  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
76anbi2i 676 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  \/  x  e.  C )
) )
8 elun 3475 . . 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 269 . 2  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  x  e.  ( ( A  i^i  B )  u.  ( A  i^i  C ) ) )
109ineqri 3521 1  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3305    i^i cin 3306
This theorem is referenced by:  indir  3576  difindi  3582  undisj2  3667  disjssun  3672  difdifdir  3702  disjpr2  3857  diftpsn3  3924  resundi  5146  fresaun  5600  elfiun  7421  unxpwdom  7541  kmlem2  8015  cdainf  8056  ackbij1lem1  8084  ackbij1lem2  8085  ssxr  9129  incexclem  12599  bitsinv1  12937  bitsinvp1  12944  bitsres  12968  paste  17341  unmbl  19415  ovolioo  19445  uniioombllem4  19461  volcn  19481  ellimc2  19747  lhop2  19882  ex-in  21716
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 2411
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-v 2945  df-un 3312  df-in 3314
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