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Theorem dfin3 3315
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 3222 . 2  |-  ( _V 
\  ( _V  \ 
( A  \  ( _V  \  B ) ) ) )  =  ( A  \  ( _V 
\  B ) )
2 dfun2 3311 . . . 4  |-  ( ( _V  \  A )  u.  ( _V  \  B ) )  =  ( _V  \  (
( _V  \  ( _V  \  A ) ) 
\  ( _V  \  B ) ) )
3 ddif 3222 . . . . . 6  |-  ( _V 
\  ( _V  \  A ) )  =  A
43difeq1i 3207 . . . . 5  |-  ( ( _V  \  ( _V 
\  A ) ) 
\  ( _V  \  B ) )  =  ( A  \  ( _V  \  B ) )
54difeq2i 3208 . . . 4  |-  ( _V 
\  ( ( _V 
\  ( _V  \  A ) )  \ 
( _V  \  B
) ) )  =  ( _V  \  ( A  \  ( _V  \  B ) ) )
62, 5eqtri 2273 . . 3  |-  ( ( _V  \  A )  u.  ( _V  \  B ) )  =  ( _V  \  ( A  \  ( _V  \  B ) ) )
76difeq2i 3208 . 2  |-  ( _V 
\  ( ( _V 
\  A )  u.  ( _V  \  B
) ) )  =  ( _V  \  ( _V  \  ( A  \ 
( _V  \  B
) ) ) )
8 dfin2 3312 . 2  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )
91, 7, 83eqtr4ri 2284 1  |-  ( A  i^i  B )  =  ( _V  \  (
( _V  \  A
)  u.  ( _V 
\  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   _Vcvv 2727    \ cdif 3075    u. cun 3076    i^i cin 3077
This theorem is referenced by:  difindi  3330
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085
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