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Theorem unssin 3236
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 845 . . . . 5  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  ( -.  x  e.  A  /\  -.  x  e.  B ) )
2 eldifn 3121 . . . . . 6  |-  ( x  e.  ( _V  \  A )  ->  -.  x  e.  A )
3 eldifn 3121 . . . . . 6  |-  ( x  e.  ( _V  \  B )  ->  -.  x  e.  B )
42, 3anim12i 331 . . . . 5  |-  ( ( x  e.  ( _V 
\  A )  /\  x  e.  ( _V  \  B ) )  -> 
( -.  x  e.  A  /\  -.  x  e.  B ) )
51, 4nsyl 593 . . . 4  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  ( x  e.  ( _V  \  A
)  /\  x  e.  ( _V  \  B ) ) )
6 elin 3181 . . . 4  |-  ( x  e.  ( ( _V 
\  A )  i^i  ( _V  \  B
) )  <->  ( x  e.  ( _V  \  A
)  /\  x  e.  ( _V  \  B ) ) )
75, 6sylnibr 637 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  x  e.  ( ( _V  \  A
)  i^i  ( _V  \  B ) ) )
8 elun 3139 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
9 vex 2622 . . . 4  |-  x  e. 
_V
10 eldif 3006 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  i^i  ( _V  \  B ) ) )  <->  ( x  e. 
_V  /\  -.  x  e.  ( ( _V  \  A )  i^i  ( _V  \  B ) ) ) )
119, 10mpbiran 886 . . 3  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  i^i  ( _V  \  B ) ) )  <->  -.  x  e.  ( ( _V  \  A )  i^i  ( _V  \  B ) ) )
127, 8, 113imtr4i 199 . 2  |-  ( x  e.  ( A  u.  B )  ->  x  e.  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) ) )
1312ssriv 3027 1  |-  ( A  u.  B )  C_  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  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    i^i cin 2996    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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