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Theorem unssdif 3281
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 2663 . . . . . . . 8  |-  x  e. 
_V
2 eldif 3050 . . . . . . . 8  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
31, 2mpbiran 909 . . . . . . 7  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
43anbi1i 453 . . . . . 6  |-  ( ( x  e.  ( _V 
\  A )  /\  -.  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3050 . . . . . 6  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  ( x  e.  ( _V  \  A
)  /\  -.  x  e.  B ) )
6 ioran 726 . . . . . 6  |-  ( -.  ( x  e.  A  \/  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
74, 5, 63bitr4i 211 . . . . 5  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  -.  (
x  e.  A  \/  x  e.  B )
)
87biimpi 119 . . . 4  |-  ( x  e.  ( ( _V 
\  A )  \  B )  ->  -.  ( x  e.  A  \/  x  e.  B
) )
98con2i 601 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  x  e.  ( ( _V  \  A
)  \  B )
)
10 elun 3187 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
11 eldif 3050 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  ( x  e.  _V  /\  -.  x  e.  ( ( _V  \  A )  \  B
) ) )
121, 11mpbiran 909 . . 3  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  -.  x  e.  ( ( _V  \  A )  \  B
) )
139, 10, 123imtr4i 200 . 2  |-  ( x  e.  ( A  u.  B )  ->  x  e.  ( _V  \  (
( _V  \  A
)  \  B )
) )
1413ssriv 3071 1  |-  ( A  u.  B )  C_  ( _V  \  (
( _V  \  A
)  \  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 682    e. wcel 1465   _Vcvv 2660    \ cdif 3038    u. cun 3039    C_ wss 3041
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
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