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Theorem unssin 3366
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 776 . . . . 5  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  ( -.  x  e.  A  /\  -.  x  e.  B ) )
2 eldifn 3250 . . . . . 6  |-  ( x  e.  ( _V  \  A )  ->  -.  x  e.  A )
3 eldifn 3250 . . . . . 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 623 . . . 4  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  ( x  e.  ( _V  \  A
)  /\  x  e.  ( _V  \  B ) ) )
6 elin 3310 . . . 4  |-  ( x  e.  ( ( _V 
\  A )  i^i  ( _V  \  B
) )  <->  ( x  e.  ( _V  \  A
)  /\  x  e.  ( _V  \  B ) ) )
75, 6sylnibr 672 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  ->  -.  x  e.  ( ( _V  \  A
)  i^i  ( _V  \  B ) ) )
8 elun 3268 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
9 vex 2733 . . . 4  |-  x  e. 
_V
10 eldif 3130 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  i^i  ( _V  \  B ) ) )  <->  ( x  e. 
_V  /\  -.  x  e.  ( ( _V  \  A )  i^i  ( _V  \  B ) ) ) )
119, 10mpbiran 935 . . 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 3151 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 703    e. wcel 2141   _Vcvv 2730    \ cdif 3118    u. cun 3119    i^i cin 3120    C_ wss 3121
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134
This theorem is referenced by: (None)
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