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Theorem difdifdirss 3478
Description: Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
difdifdirss  |-  ( ( A  \  B ) 
\  C )  C_  ( ( A  \  C )  \  ( B  \  C ) )

Proof of Theorem difdifdirss
StepHypRef Expression
1 dif32 3370 . . . . 5  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
2 invdif 3349 . . . . 5  |-  ( ( A  \  C )  i^i  ( _V  \  B ) )  =  ( ( A  \  C )  \  B
)
31, 2eqtr4i 2181 . . . 4  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( _V  \  B ) )
4 un0 3427 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( A  \  C
)  i^i  ( _V  \  B ) )
53, 4eqtr4i 2181 . . 3  |-  ( ( A  \  B ) 
\  C )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
6 indi 3354 . . . 4  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (
( A  \  C
)  i^i  C )
)
7 disjdif 3466 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  (/)
8 incom 3299 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  C
)
97, 8eqtr3i 2180 . . . . 5  |-  (/)  =  ( ( A  \  C
)  i^i  C )
109uneq2i 3258 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( ( A  \  C )  i^i  ( _V  \  B ) )  u.  ( ( A 
\  C )  i^i 
C ) )
116, 10eqtr4i 2181 . . 3  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
125, 11eqtr4i 2181 . 2  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  (
( _V  \  B
)  u.  C ) )
13 ddifss 3345 . . . . . 6  |-  C  C_  ( _V  \  ( _V  \  C ) )
14 unss2 3278 . . . . . 6  |-  ( C 
C_  ( _V  \ 
( _V  \  C
) )  ->  (
( _V  \  B
)  u.  C ) 
C_  ( ( _V 
\  B )  u.  ( _V  \  ( _V  \  C ) ) ) )
1513, 14ax-mp 5 . . . . 5  |-  ( ( _V  \  B )  u.  C )  C_  ( ( _V  \  B )  u.  ( _V  \  ( _V  \  C ) ) )
16 indmss 3366 . . . . . 6  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  C_  ( _V  \  ( B  i^i  ( _V  \  C ) ) )
17 invdif 3349 . . . . . . 7  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
1817difeq2i 3222 . . . . . 6  |-  ( _V 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( _V  \  ( B 
\  C ) )
1916, 18sseqtri 3162 . . . . 5  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  C_  ( _V  \  ( B  \  C ) )
2015, 19sstri 3137 . . . 4  |-  ( ( _V  \  B )  u.  C )  C_  ( _V  \  ( B  \  C ) )
21 sslin 3333 . . . 4  |-  ( ( ( _V  \  B
)  u.  C ) 
C_  ( _V  \ 
( B  \  C
) )  ->  (
( A  \  C
)  i^i  ( ( _V  \  B )  u.  C ) )  C_  ( ( A  \  C )  i^i  ( _V  \  ( B  \  C ) ) ) )
2220, 21ax-mp 5 . . 3  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  C_  ( ( A  \  C )  i^i  ( _V  \  ( B  \  C ) ) )
23 invdif 3349 . . 3  |-  ( ( A  \  C )  i^i  ( _V  \ 
( B  \  C
) ) )  =  ( ( A  \  C )  \  ( B  \  C ) )
2422, 23sseqtri 3162 . 2  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  C_  ( ( A  \  C )  \  ( B  \  C ) )
2512, 24eqsstri 3160 1  |-  ( ( A  \  B ) 
\  C )  C_  ( ( A  \  C )  \  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2712    \ cdif 3099    u. cun 3100    i^i cin 3101    C_ wss 3102   (/)c0 3394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395
This theorem is referenced by: (None)
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