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Theorem dfin3 3421
Description: Intersection defined in terms of union (De Morgan'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 3321 . 2  |-  ( _V 
\  ( _V  \ 
( A  \  ( _V  \  B ) ) ) )  =  ( A  \  ( _V 
\  B ) )
2 dfun2 3417 . . . 4  |-  ( ( _V  \  A )  u.  ( _V  \  B ) )  =  ( _V  \  (
( _V  \  ( _V  \  A ) ) 
\  ( _V  \  B ) ) )
3 ddif 3321 . . . . . 6  |-  ( _V 
\  ( _V  \  A ) )  =  A
43difeq1i 3303 . . . . 5  |-  ( ( _V  \  ( _V 
\  A ) ) 
\  ( _V  \  B ) )  =  ( A  \  ( _V  \  B ) )
54difeq2i 3304 . . . 4  |-  ( _V 
\  ( ( _V 
\  ( _V  \  A ) )  \ 
( _V  \  B
) ) )  =  ( _V  \  ( A  \  ( _V  \  B ) ) )
62, 5eqtri 2316 . . 3  |-  ( ( _V  \  A )  u.  ( _V  \  B ) )  =  ( _V  \  ( A  \  ( _V  \  B ) ) )
76difeq2i 3304 . 2  |-  ( _V 
\  ( ( _V 
\  A )  u.  ( _V  \  B
) ) )  =  ( _V  \  ( _V  \  ( A  \ 
( _V  \  B
) ) ) )
8 dfin2 3418 . 2  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )
91, 7, 83eqtr4ri 2327 1  |-  ( A  i^i  B )  =  ( _V  \  (
( _V  \  A
)  u.  ( _V 
\  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164
This theorem is referenced by:  difindi  3436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172
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