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Theorem inssdif 3173
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 3068 . . . 4  |-  ( x  e.  B  ->  -.  x  e.  ( _V  \  B ) )
21anim2i 324 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  ->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B ) ) )
3 elin 3126 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
4 eldif 2927 . . 3  |-  ( x  e.  ( A  \ 
( _V  \  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B
) ) )
52, 3, 43imtr4i 190 . 2  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  ( A  \  ( _V  \  B ) ) )
65ssriv 2949 1  |-  ( A  i^i  B )  C_  ( A  \  ( _V  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 97    e. wcel 1393   _Vcvv 2557    \ cdif 2914    i^i cin 2916    C_ wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931
This theorem is referenced by:  difdif2ss  3194
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