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Theorem difun1 3428
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 3379 . . . 4  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( _V  \  C ) )  =  ( A  i^i  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
2 invdif 3410 . . . 4  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  B ) )  \  C )
31, 2eqtr3i 2305 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  \  C )
4 undm 3426 . . . . 5  |-  ( _V 
\  ( B  u.  C ) )  =  ( ( _V  \  B )  i^i  ( _V  \  C ) )
54ineq2i 3367 . . . 4  |-  ( A  i^i  ( _V  \ 
( B  u.  C
) ) )  =  ( A  i^i  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
6 invdif 3410 . . . 4  |-  ( A  i^i  ( _V  \ 
( B  u.  C
) ) )  =  ( A  \  ( B  u.  C )
)
75, 6eqtr3i 2305 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( A  \  ( B  u.  C )
)
83, 7eqtr3i 2305 . 2  |-  ( ( A  i^i  ( _V 
\  B ) ) 
\  C )  =  ( A  \  ( B  u.  C )
)
9 invdif 3410 . . 3  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
109difeq1i 3290 . 2  |-  ( ( A  i^i  ( _V 
\  B ) ) 
\  C )  =  ( ( A  \  B )  \  C
)
118, 10eqtr3i 2305 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151
This theorem is referenced by:  dif32  3431  difabs  3432  infdiffi  7358  mreexexlem4d  13549  nulmbl2  18894  unmbl  18895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159
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