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Theorem inssdif 3371
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 3259 . . . 4  |-  ( x  e.  B  ->  -.  x  e.  ( _V  \  B ) )
21anim2i 342 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  ->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B ) ) )
3 elin 3318 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
4 eldif 3138 . . 3  |-  ( x  e.  ( A  \ 
( _V  \  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B
) ) )
52, 3, 43imtr4i 201 . 2  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  ( A  \  ( _V  \  B ) ) )
65ssriv 3159 1  |-  ( A  i^i  B )  C_  ( A  \  ( _V  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    e. wcel 2148   _Vcvv 2737    \ cdif 3126    i^i cin 3128    C_ wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142
This theorem is referenced by:  difdif2ss  3392
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