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Theorem difindiss 3331
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 3218 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( A  \  C
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( A  \  C
) ) )
2 eldif 3081 . . . . . . 7  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
3 eldif 3081 . . . . . . 7  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
42, 3orbi12i 754 . . . . . 6  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( A 
\  C ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  A  /\  -.  x  e.  C
) ) )
5 andi 808 . . . . . 6  |-  ( ( x  e.  A  /\  ( -.  x  e.  B  \/  -.  x  e.  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  \/  ( x  e.  A  /\  -.  x  e.  C )
) )
64, 5bitr4i 186 . . . . 5  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( A 
\  C ) )  <-> 
( x  e.  A  /\  ( -.  x  e.  B  \/  -.  x  e.  C ) ) )
7 pm3.14 743 . . . . . 6  |-  ( ( -.  x  e.  B  \/  -.  x  e.  C
)  ->  -.  (
x  e.  B  /\  x  e.  C )
)
87anim2i 340 . . . . 5  |-  ( ( x  e.  A  /\  ( -.  x  e.  B  \/  -.  x  e.  C ) )  -> 
( x  e.  A  /\  -.  ( x  e.  B  /\  x  e.  C ) ) )
96, 8sylbi 120 . . . 4  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( A 
\  C ) )  ->  ( x  e.  A  /\  -.  (
x  e.  B  /\  x  e.  C )
) )
10 eldif 3081 . . . . 5  |-  ( x  e.  ( A  \ 
( B  i^i  C
) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  i^i  C
) ) )
11 elin 3260 . . . . . . 7  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
1211notbii 658 . . . . . 6  |-  ( -.  x  e.  ( B  i^i  C )  <->  -.  (
x  e.  B  /\  x  e.  C )
)
1312anbi2i 453 . . . . 5  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  i^i  C ) )  <-> 
( x  e.  A  /\  -.  ( x  e.  B  /\  x  e.  C ) ) )
1410, 13bitr2i 184 . . . 4  |-  ( ( x  e.  A  /\  -.  ( x  e.  B  /\  x  e.  C
) )  <->  x  e.  ( A  \  ( B  i^i  C ) ) )
159, 14sylib 121 . . 3  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( A 
\  C ) )  ->  x  e.  ( A  \  ( B  i^i  C ) ) )
161, 15sylbi 120 . 2  |-  ( x  e.  ( ( A 
\  B )  u.  ( A  \  C
) )  ->  x  e.  ( A  \  ( B  i^i  C ) ) )
1716ssriv 3102 1  |-  ( ( A  \  B )  u.  ( A  \  C ) )  C_  ( A  \  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 698    e. wcel 1481    \ cdif 3069    u. cun 3070    i^i cin 3071    C_ wss 3072
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085
This theorem is referenced by:  difdif2ss  3334  indmss  3336
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