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

Theorem difundi 3750
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 3736 . . 3  |-  ( B  u.  C )  =  ( _V  \  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
21difeq2i 3619 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) ) )
3 inindi 3715 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  i^i  ( A  i^i  ( _V  \  C ) ) )
4 dfin2 3734 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) ) )
5 invdif 3739 . . . 4  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
6 invdif 3739 . . . 4  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
75, 6ineq12i 3698 . . 3  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( A  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  i^i  ( A  \  C ) )
83, 4, 73eqtr3i 2504 . 2  |-  ( A 
\  ( _V  \ 
( ( _V  \  B )  i^i  ( _V  \  C ) ) ) )  =  ( ( A  \  B
)  i^i  ( A  \  C ) )
92, 8eqtri 2496 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483
This theorem is referenced by:  undm  3756  uncld  19305  inmbl  21684  clsun  29721
  Copyright terms: Public domain W3C validator