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Theorem indi 3751
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 867 . . . 4  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( (
x  e.  A  /\  x  e.  B )  \/  ( x  e.  A  /\  x  e.  C
) ) )
2 elin 3683 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3 elin 3683 . . . . 5  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
42, 3orbi12i 521 . . . 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 252 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  ( A  i^i  B
)  \/  x  e.  ( A  i^i  C
) ) )
6 elun 3641 . . . 4  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
76anbi2i 694 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  \/  x  e.  C )
) )
8 elun 3641 . . 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 277 . 2  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  x  e.  ( ( A  i^i  B )  u.  ( A  i^i  C ) ) )
109ineqri 3688 1  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    u. cun 3469    i^i cin 3470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-in 3478
This theorem is referenced by:  indir  3753  difindi  3759  undisj2  3882  disjssun  3887  difdifdir  3918  disjpr2  4094  diftpsn3  4170  resundi  5297  fresaun  5762  elfiun  7908  unxpwdom  8033  kmlem2  8548  cdainf  8589  ackbij1lem1  8617  ackbij1lem2  8618  ssxr  9671  incexclem  13659  bitsinv1  14103  bitsinvp1  14110  bitsres  14134  paste  19921  unmbl  22073  ovolioo  22103  uniioombllem4  22120  volcn  22140  ellimc2  22406  lhop2  22541  ex-in  25272  eulerpartgbij  28486  asindmre  30264
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