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Theorem inssdif 3312
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 3200 . . . 4  |-  ( x  e.  B  ->  -.  x  e.  ( _V  \  B ) )
21anim2i 339 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  ->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B ) ) )
3 elin 3259 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
4 eldif 3080 . . 3  |-  ( x  e.  ( A  \ 
( _V  \  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B
) ) )
52, 3, 43imtr4i 200 . 2  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  ( A  \  ( _V  \  B ) ) )
65ssriv 3101 1  |-  ( A  i^i  B )  C_  ( A  \  ( _V  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    e. wcel 1480   _Vcvv 2686    \ cdif 3068    i^i cin 3070    C_ wss 3071
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084
This theorem is referenced by:  difdif2ss  3333
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