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Theorem difun1 3441
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 3392 . . . 4  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( _V  \  C ) )  =  ( A  i^i  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
2 invdif 3423 . . . 4  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  B ) )  \  C )
31, 2eqtr3i 2318 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  \  C )
4 undm 3439 . . . . 5  |-  ( _V 
\  ( B  u.  C ) )  =  ( ( _V  \  B )  i^i  ( _V  \  C ) )
54ineq2i 3380 . . . 4  |-  ( A  i^i  ( _V  \ 
( B  u.  C
) ) )  =  ( A  i^i  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
6 invdif 3423 . . . 4  |-  ( A  i^i  ( _V  \ 
( B  u.  C
) ) )  =  ( A  \  ( B  u.  C )
)
75, 6eqtr3i 2318 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( A  \  ( B  u.  C )
)
83, 7eqtr3i 2318 . 2  |-  ( ( A  i^i  ( _V 
\  B ) ) 
\  C )  =  ( A  \  ( B  u.  C )
)
9 invdif 3423 . . 3  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
109difeq1i 3303 . 2  |-  ( ( A  i^i  ( _V 
\  B ) ) 
\  C )  =  ( ( A  \  B )  \  C
)
118, 10eqtr3i 2318 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164
This theorem is referenced by:  dif32  3444  difabs  3445  infdiffi  7374  mreexexlem4d  13565  nulmbl2  18910  unmbl  18911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172
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