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Theorem unssdif 3382
Description: Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
unssdif  |-  ( A  u.  B )  C_  ( _V  \  (
( _V  \  A
)  \  B )
)

Proof of Theorem unssdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2752 . . . . . . . 8  |-  x  e. 
_V
2 eldif 3150 . . . . . . . 8  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
31, 2mpbiran 941 . . . . . . 7  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
43anbi1i 458 . . . . . 6  |-  ( ( x  e.  ( _V 
\  A )  /\  -.  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3150 . . . . . 6  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  ( x  e.  ( _V  \  A
)  /\  -.  x  e.  B ) )
6 ioran 753 . . . . . 6  |-  ( -.  ( x  e.  A  \/  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
74, 5, 63bitr4i 212 . . . . 5  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  -.  (
x  e.  A  \/  x  e.  B )
)
87biimpi 120 . . . 4  |-  ( x  e.  ( ( _V 
\  A )  \  B )  ->  -.  ( x  e.  A  \/  x  e.  B
) )
98con2i 628 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  x  e.  ( ( _V  \  A
)  \  B )
)
10 elun 3288 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
11 eldif 3150 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  ( x  e.  _V  /\  -.  x  e.  ( ( _V  \  A )  \  B
) ) )
121, 11mpbiran 941 . . 3  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  -.  x  e.  ( ( _V  \  A )  \  B
) )
139, 10, 123imtr4i 201 . 2  |-  ( x  e.  ( A  u.  B )  ->  x  e.  ( _V  \  (
( _V  \  A
)  \  B )
) )
1413ssriv 3171 1  |-  ( A  u.  B )  C_  ( _V  \  (
( _V  \  A
)  \  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    \/ wo 709    e. wcel 2158   _Vcvv 2749    \ cdif 3138    u. cun 3139    C_ wss 3141
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154
This theorem is referenced by: (None)
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