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

Theorem difundi 3701
Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundi  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )

Proof of Theorem difundi
StepHypRef Expression
1 dfun3 3687 . . 3  |-  ( B  u.  C )  =  ( _V  \  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
21difeq2i 3557 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) ) )
3 inindi 3655 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  i^i  ( A  i^i  ( _V  \  C ) ) )
4 dfin2 3685 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) ) )
5 invdif 3690 . . . 4  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
6 invdif 3690 . . . 4  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
75, 6ineq12i 3638 . . 3  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( A  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  i^i  ( A  \  C ) )
83, 4, 73eqtr3i 2439 . 2  |-  ( A 
\  ( _V  \ 
( ( _V  \  B )  i^i  ( _V  \  C ) ) ) )  =  ( ( A  \  B
)  i^i  ( A  \  C ) )
92, 8eqtri 2431 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   _Vcvv 3058    \ cdif 3410    u. cun 3411    i^i cin 3412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420
This theorem is referenced by:  undm  3707  uncld  19726  inmbl  22136  difuncomp  27728  clsun  30544
  Copyright terms: Public domain W3C validator