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Theorem difindiss 3251
Description: Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
Assertion
Ref Expression
difindiss  |-  ( ( A  \  B )  u.  ( A  \  C ) )  C_  ( A  \  ( B  i^i  C ) )

Proof of Theorem difindiss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elun 3139 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( A  \  C
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( A  \  C
) ) )
2 eldif 3006 . . . . . . 7  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
3 eldif 3006 . . . . . . 7  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
42, 3orbi12i 716 . . . . . 6  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( A 
\  C ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  A  /\  -.  x  e.  C
) ) )
5 andi 767 . . . . . 6  |-  ( ( x  e.  A  /\  ( -.  x  e.  B  \/  -.  x  e.  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  \/  ( x  e.  A  /\  -.  x  e.  C )
) )
64, 5bitr4i 185 . . . . 5  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( A 
\  C ) )  <-> 
( x  e.  A  /\  ( -.  x  e.  B  \/  -.  x  e.  C ) ) )
7 pm3.14 705 . . . . . 6  |-  ( ( -.  x  e.  B  \/  -.  x  e.  C
)  ->  -.  (
x  e.  B  /\  x  e.  C )
)
87anim2i 334 . . . . 5  |-  ( ( x  e.  A  /\  ( -.  x  e.  B  \/  -.  x  e.  C ) )  -> 
( x  e.  A  /\  -.  ( x  e.  B  /\  x  e.  C ) ) )
96, 8sylbi 119 . . . 4  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( A 
\  C ) )  ->  ( x  e.  A  /\  -.  (
x  e.  B  /\  x  e.  C )
) )
10 eldif 3006 . . . . 5  |-  ( x  e.  ( A  \ 
( B  i^i  C
) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  i^i  C
) ) )
11 elin 3181 . . . . . . 7  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
1211notbii 629 . . . . . 6  |-  ( -.  x  e.  ( B  i^i  C )  <->  -.  (
x  e.  B  /\  x  e.  C )
)
1312anbi2i 445 . . . . 5  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  i^i  C ) )  <-> 
( x  e.  A  /\  -.  ( x  e.  B  /\  x  e.  C ) ) )
1410, 13bitr2i 183 . . . 4  |-  ( ( x  e.  A  /\  -.  ( x  e.  B  /\  x  e.  C
) )  <->  x  e.  ( A  \  ( B  i^i  C ) ) )
159, 14sylib 120 . . 3  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( A 
\  C ) )  ->  x  e.  ( A  \  ( B  i^i  C ) ) )
161, 15sylbi 119 . 2  |-  ( x  e.  ( ( A 
\  B )  u.  ( A  \  C
) )  ->  x  e.  ( A  \  ( B  i^i  C ) ) )
1716ssriv 3027 1  |-  ( ( A  \  B )  u.  ( A  \  C ) )  C_  ( A  \  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    \/ wo 664    e. wcel 1438    \ cdif 2994    u. cun 2995    i^i cin 2996    C_ wss 2997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010
This theorem is referenced by:  difdif2ss  3254  indmss  3256
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