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Theorem dfin3 3383
Description: Intersection defined in terms of union (DeMorgan's law. Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
dfin3  |-  ( A  i^i  B )  =  ( _V  \  (
( _V  \  A
)  u.  ( _V 
\  B ) ) )

Proof of Theorem dfin3
StepHypRef Expression
1 ddif 3283 . 2  |-  ( _V 
\  ( _V  \ 
( A  \  ( _V  \  B ) ) ) )  =  ( A  \  ( _V 
\  B ) )
2 dfun2 3379 . . . 4  |-  ( ( _V  \  A )  u.  ( _V  \  B ) )  =  ( _V  \  (
( _V  \  ( _V  \  A ) ) 
\  ( _V  \  B ) ) )
3 ddif 3283 . . . . . 6  |-  ( _V 
\  ( _V  \  A ) )  =  A
43difeq1i 3265 . . . . 5  |-  ( ( _V  \  ( _V 
\  A ) ) 
\  ( _V  \  B ) )  =  ( A  \  ( _V  \  B ) )
54difeq2i 3266 . . . 4  |-  ( _V 
\  ( ( _V 
\  ( _V  \  A ) )  \ 
( _V  \  B
) ) )  =  ( _V  \  ( A  \  ( _V  \  B ) ) )
62, 5eqtri 2278 . . 3  |-  ( ( _V  \  A )  u.  ( _V  \  B ) )  =  ( _V  \  ( A  \  ( _V  \  B ) ) )
76difeq2i 3266 . 2  |-  ( _V 
\  ( ( _V 
\  A )  u.  ( _V  \  B
) ) )  =  ( _V  \  ( _V  \  ( A  \ 
( _V  \  B
) ) ) )
8 dfin2 3380 . 2  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )
91, 7, 83eqtr4ri 2289 1  |-  ( A  i^i  B )  =  ( _V  \  (
( _V  \  A
)  u.  ( _V 
\  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   _Vcvv 2763    \ cdif 3124    u. cun 3125    i^i cin 3126
This theorem is referenced by:  difindi  3398
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134
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