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Theorem difdifdirss 3519
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 3410 . . . . 5  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
2 invdif 3389 . . . . 5  |-  ( ( A  \  C )  i^i  ( _V  \  B ) )  =  ( ( A  \  C )  \  B
)
31, 2eqtr4i 2211 . . . 4  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( _V  \  B ) )
4 un0 3468 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( A  \  C
)  i^i  ( _V  \  B ) )
53, 4eqtr4i 2211 . . 3  |-  ( ( A  \  B ) 
\  C )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
6 indi 3394 . . . 4  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (
( A  \  C
)  i^i  C )
)
7 disjdif 3507 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  (/)
8 incom 3339 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  C
)
97, 8eqtr3i 2210 . . . . 5  |-  (/)  =  ( ( A  \  C
)  i^i  C )
109uneq2i 3298 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( ( A  \  C )  i^i  ( _V  \  B ) )  u.  ( ( A 
\  C )  i^i 
C ) )
116, 10eqtr4i 2211 . . 3  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
125, 11eqtr4i 2211 . 2  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  (
( _V  \  B
)  u.  C ) )
13 ddifss 3385 . . . . . 6  |-  C  C_  ( _V  \  ( _V  \  C ) )
14 unss2 3318 . . . . . 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 3406 . . . . . 6  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  C_  ( _V  \  ( B  i^i  ( _V  \  C ) ) )
17 invdif 3389 . . . . . . 7  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
1817difeq2i 3262 . . . . . 6  |-  ( _V 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( _V  \  ( B 
\  C ) )
1916, 18sseqtri 3201 . . . . 5  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  C_  ( _V  \  ( B  \  C ) )
2015, 19sstri 3176 . . . 4  |-  ( ( _V  \  B )  u.  C )  C_  ( _V  \  ( B  \  C ) )
21 sslin 3373 . . . 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 3389 . . 3  |-  ( ( A  \  C )  i^i  ( _V  \ 
( B  \  C
) ) )  =  ( ( A  \  C )  \  ( B  \  C ) )
2422, 23sseqtri 3201 . 2  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  C_  ( ( A  \  C )  \  ( B  \  C ) )
2512, 24eqsstri 3199 1  |-  ( ( A  \  B ) 
\  C )  C_  ( ( A  \  C )  \  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2749    \ cdif 3138    u. cun 3139    i^i cin 3140    C_ wss 3141   (/)c0 3434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435
This theorem is referenced by: (None)
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