ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  unssdif Unicode version

Theorem unssdif 3258
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 2644 . . . . . . . 8  |-  x  e. 
_V
2 eldif 3030 . . . . . . . 8  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
31, 2mpbiran 892 . . . . . . 7  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
43anbi1i 449 . . . . . 6  |-  ( ( x  e.  ( _V 
\  A )  /\  -.  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3030 . . . . . 6  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  ( x  e.  ( _V  \  A
)  /\  -.  x  e.  B ) )
6 ioran 710 . . . . . 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 597 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  x  e.  ( ( _V  \  A
)  \  B )
)
10 elun 3164 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
11 eldif 3030 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  ( x  e.  _V  /\  -.  x  e.  ( ( _V  \  A )  \  B
) ) )
121, 11mpbiran 892 . . 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 3051 1  |-  ( A  u.  B )  C_  ( _V  \  (
( _V  \  A
)  \  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 670    e. wcel 1448   _Vcvv 2641    \ cdif 3018    u. cun 3019    C_ wss 3021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator