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Theorem difun1 3544
Description: A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
Assertion
Ref Expression
difun1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)

Proof of Theorem difun1
StepHypRef Expression
1 inass 3494 . . . 4  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( _V  \  C ) )  =  ( A  i^i  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
2 invdif 3525 . . . 4  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  B ) )  \  C )
31, 2eqtr3i 2409 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  \  C )
4 undm 3542 . . . . 5  |-  ( _V 
\  ( B  u.  C ) )  =  ( ( _V  \  B )  i^i  ( _V  \  C ) )
54ineq2i 3482 . . . 4  |-  ( A  i^i  ( _V  \ 
( B  u.  C
) ) )  =  ( A  i^i  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
6 invdif 3525 . . . 4  |-  ( A  i^i  ( _V  \ 
( B  u.  C
) ) )  =  ( A  \  ( B  u.  C )
)
75, 6eqtr3i 2409 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( A  \  ( B  u.  C )
)
83, 7eqtr3i 2409 . 2  |-  ( ( A  i^i  ( _V 
\  B ) ) 
\  C )  =  ( A  \  ( B  u.  C )
)
9 invdif 3525 . . 3  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
109difeq1i 3404 . 2  |-  ( ( A  i^i  ( _V 
\  B ) ) 
\  C )  =  ( ( A  \  B )  \  C
)
118, 10eqtr3i 2409 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2899    \ cdif 3260    u. cun 3261    i^i cin 3262
This theorem is referenced by:  dif32  3547  difabs  3548  infdiffi  7545  mreexexlem4d  13799  nulmbl2  19298  unmbl  19299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270
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