MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difindi Unicode version

Theorem difindi 3499
Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difindi  |-  ( A 
\  ( B  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )

Proof of Theorem difindi
StepHypRef Expression
1 dfin3 3484 . . 3  |-  ( B  i^i  C )  =  ( _V  \  (
( _V  \  B
)  u.  ( _V 
\  C ) ) )
21difeq2i 3367 . 2  |-  ( A 
\  ( B  i^i  C ) )  =  ( A  \  ( _V 
\  ( ( _V 
\  B )  u.  ( _V  \  C
) ) ) )
3 indi 3491 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  u.  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  u.  ( A  i^i  ( _V  \  C ) ) )
4 dfin2 3481 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  u.  ( _V  \  C
) ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  u.  ( _V  \  C
) ) ) )
5 invdif 3486 . . . 4  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
6 invdif 3486 . . . 4  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
75, 6uneq12i 3403 . . 3  |-  ( ( A  i^i  ( _V 
\  B ) )  u.  ( A  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
83, 4, 73eqtr3i 2386 . 2  |-  ( A 
\  ( _V  \ 
( ( _V  \  B )  u.  ( _V  \  C ) ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
92, 8eqtri 2378 1  |-  ( A 
\  ( B  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1642   _Vcvv 2864    \ cdif 3225    u. cun 3226    i^i cin 3227
This theorem is referenced by:  indm  3503  dprddisj2  15373  fctop  16847  cctop  16849  mretopd  16935  restcld  17009  cfinfil  17690  csdfil  17691  difdif2  23197  fndifnfp  26079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235
  Copyright terms: Public domain W3C validator