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Theorem inssdif 3216
Description: Intersection of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
inssdif  |-  ( A  i^i  B )  C_  ( A  \  ( _V  \  B ) )

Proof of Theorem inssdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elndif 3106 . . . 4  |-  ( x  e.  B  ->  -.  x  e.  ( _V  \  B ) )
21anim2i 334 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  ->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B ) ) )
3 elin 3165 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
4 eldif 2991 . . 3  |-  ( x  e.  ( A  \ 
( _V  \  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B
) ) )
52, 3, 43imtr4i 199 . 2  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  ( A  \  ( _V  \  B ) ) )
65ssriv 3012 1  |-  ( A  i^i  B )  C_  ( A  \  ( _V  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    e. wcel 1434   _Vcvv 2610    \ cdif 2979    i^i cin 2981    C_ wss 2982
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 2984  df-in 2988  df-ss 2995
This theorem is referenced by:  difdif2ss  3237
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