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Theorem unssin 3347
Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
Assertion
Ref Expression
unssin  |-  ( A  u.  B )  C_  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )

Proof of Theorem unssin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oranim 771 . . . . 5  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  ( -.  x  e.  A  /\  -.  x  e.  B ) )
2 eldifn 3231 . . . . . 6  |-  ( x  e.  ( _V  \  A )  ->  -.  x  e.  A )
3 eldifn 3231 . . . . . 6  |-  ( x  e.  ( _V  \  B )  ->  -.  x  e.  B )
42, 3anim12i 336 . . . . 5  |-  ( ( x  e.  ( _V 
\  A )  /\  x  e.  ( _V  \  B ) )  -> 
( -.  x  e.  A  /\  -.  x  e.  B ) )
51, 4nsyl 618 . . . 4  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  ( x  e.  ( _V  \  A
)  /\  x  e.  ( _V  \  B ) ) )
6 elin 3291 . . . 4  |-  ( x  e.  ( ( _V 
\  A )  i^i  ( _V  \  B
) )  <->  ( x  e.  ( _V  \  A
)  /\  x  e.  ( _V  \  B ) ) )
75, 6sylnibr 667 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  x  e.  ( ( _V  \  A
)  i^i  ( _V  \  B ) ) )
8 elun 3249 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
9 vex 2715 . . . 4  |-  x  e. 
_V
10 eldif 3111 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  i^i  ( _V  \  B ) ) )  <->  ( x  e. 
_V  /\  -.  x  e.  ( ( _V  \  A )  i^i  ( _V  \  B ) ) ) )
119, 10mpbiran 925 . . 3  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  i^i  ( _V  \  B ) ) )  <->  -.  x  e.  ( ( _V  \  A )  i^i  ( _V  \  B ) ) )
127, 8, 113imtr4i 200 . 2  |-  ( x  e.  ( A  u.  B )  ->  x  e.  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) ) )
1312ssriv 3132 1  |-  ( A  u.  B )  C_  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 698    e. wcel 2128   _Vcvv 2712    \ cdif 3099    u. cun 3100    i^i cin 3101    C_ wss 3102
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115
This theorem is referenced by: (None)
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