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Theorem inssun 3284
Description: Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
Assertion
Ref Expression
inssun  |-  ( A  i^i  B )  C_  ( _V  \  (
( _V  \  A
)  u.  ( _V 
\  B ) ) )

Proof of Theorem inssun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pm3.1 726 . . . . 5  |-  ( ( x  e.  A  /\  x  e.  B )  ->  -.  ( -.  x  e.  A  \/  -.  x  e.  B )
)
2 eldifn 3167 . . . . . 6  |-  ( x  e.  ( _V  \  A )  ->  -.  x  e.  A )
3 eldifn 3167 . . . . . 6  |-  ( x  e.  ( _V  \  B )  ->  -.  x  e.  B )
42, 3orim12i 731 . . . . 5  |-  ( ( x  e.  ( _V 
\  A )  \/  x  e.  ( _V 
\  B ) )  ->  ( -.  x  e.  A  \/  -.  x  e.  B )
)
51, 4nsyl 600 . . . 4  |-  ( ( x  e.  A  /\  x  e.  B )  ->  -.  ( x  e.  ( _V  \  A
)  \/  x  e.  ( _V  \  B
) ) )
6 elun 3185 . . . 4  |-  ( x  e.  ( ( _V 
\  A )  u.  ( _V  \  B
) )  <->  ( x  e.  ( _V  \  A
)  \/  x  e.  ( _V  \  B
) ) )
75, 6sylnibr 649 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  ->  -.  x  e.  ( ( _V  \  A
)  u.  ( _V 
\  B ) ) )
8 elin 3227 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
9 vex 2661 . . . 4  |-  x  e. 
_V
10 eldif 3048 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  u.  ( _V  \  B ) ) )  <->  ( x  e. 
_V  /\  -.  x  e.  ( ( _V  \  A )  u.  ( _V  \  B ) ) ) )
119, 10mpbiran 907 . . 3  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  u.  ( _V  \  B ) ) )  <->  -.  x  e.  ( ( _V  \  A )  u.  ( _V  \  B ) ) )
127, 8, 113imtr4i 200 . 2  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  ( _V  \  (
( _V  \  A
)  u.  ( _V 
\  B ) ) ) )
1312ssriv 3069 1  |-  ( A  i^i  B )  C_  ( _V  \  (
( _V  \  A
)  u.  ( _V 
\  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 680    e. wcel 1463   _Vcvv 2658    \ cdif 3036    u. cun 3037    i^i cin 3038    C_ wss 3039
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052
This theorem is referenced by: (None)
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