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Theorem difun1 3593
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 3543 . . . 4  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( _V  \  C ) )  =  ( A  i^i  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
2 invdif 3574 . . . 4  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  B ) )  \  C )
31, 2eqtr3i 2457 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  \  C )
4 undm 3591 . . . . 5  |-  ( _V 
\  ( B  u.  C ) )  =  ( ( _V  \  B )  i^i  ( _V  \  C ) )
54ineq2i 3531 . . . 4  |-  ( A  i^i  ( _V  \ 
( B  u.  C
) ) )  =  ( A  i^i  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
6 invdif 3574 . . . 4  |-  ( A  i^i  ( _V  \ 
( B  u.  C
) ) )  =  ( A  \  ( B  u.  C )
)
75, 6eqtr3i 2457 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( A  \  ( B  u.  C )
)
83, 7eqtr3i 2457 . 2  |-  ( ( A  i^i  ( _V 
\  B ) ) 
\  C )  =  ( A  \  ( B  u.  C )
)
9 invdif 3574 . . 3  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
109difeq1i 3453 . 2  |-  ( ( A  i^i  ( _V 
\  B ) ) 
\  C )  =  ( ( A  \  B )  \  C
)
118, 10eqtr3i 2457 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311
This theorem is referenced by:  dif32  3596  difabs  3597  infdiffi  7604  mreexexlem4d  13864  nulmbl2  19423  unmbl  19424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319
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