ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difindiss Unicode version

Theorem difindiss 3235
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 3124 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( A  \  C
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( A  \  C
) ) )
2 eldif 2992 . . . . . . 7  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
3 eldif 2992 . . . . . . 7  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
42, 3orbi12i 714 . . . . . 6  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( A 
\  C ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  A  /\  -.  x  e.  C
) ) )
5 andi 765 . . . . . 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 pm3.14 703 . . . . . 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 2992 . . . . 5  |-  ( x  e.  ( A  \ 
( B  i^i  C
) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  i^i  C
) ) )
11 elin 3166 . . . . . . 7  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
1211notbii 627 . . . . . 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 3013 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 662    e. wcel 1434    \ cdif 2980    u. cun 2981    i^i cin 2982    C_ wss 2983
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996
This theorem is referenced by:  difdif2ss  3238  indmss  3240
  Copyright terms: Public domain W3C validator