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Theorem unssdif 3232
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 2622 . . . . . . . 8  |-  x  e. 
_V
2 eldif 3006 . . . . . . . 8  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
31, 2mpbiran 886 . . . . . . 7  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
43anbi1i 446 . . . . . 6  |-  ( ( x  e.  ( _V 
\  A )  /\  -.  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3006 . . . . . 6  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  ( x  e.  ( _V  \  A
)  /\  -.  x  e.  B ) )
6 ioran 704 . . . . . 6  |-  ( -.  ( x  e.  A  \/  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
74, 5, 63bitr4i 210 . . . . 5  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  -.  (
x  e.  A  \/  x  e.  B )
)
87biimpi 118 . . . 4  |-  ( x  e.  ( ( _V 
\  A )  \  B )  ->  -.  ( x  e.  A  \/  x  e.  B
) )
98con2i 592 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  x  e.  ( ( _V  \  A
)  \  B )
)
10 elun 3139 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
11 eldif 3006 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  ( x  e.  _V  /\  -.  x  e.  ( ( _V  \  A )  \  B
) ) )
121, 11mpbiran 886 . . 3  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  -.  x  e.  ( ( _V  \  A )  \  B
) )
139, 10, 123imtr4i 199 . 2  |-  ( x  e.  ( A  u.  B )  ->  x  e.  ( _V  \  (
( _V  \  A
)  \  B )
) )
1413ssriv 3027 1  |-  ( A  u.  B )  C_  ( _V  \  (
( _V  \  A
)  \  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    \/ wo 664    e. wcel 1438   _Vcvv 2619    \ cdif 2994    u. cun 2995    C_ wss 2997
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010
This theorem is referenced by: (None)
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