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Theorem dfun3 3567
Description: Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
dfun3  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )

Proof of Theorem dfun3
StepHypRef Expression
1 dfun2 3564 . 2  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  \  B )
)
2 dfin2 3565 . . . 4  |-  ( ( _V  \  A )  i^i  ( _V  \  B ) )  =  ( ( _V  \  A )  \  ( _V  \  ( _V  \  B ) ) )
3 ddif 3468 . . . . 5  |-  ( _V 
\  ( _V  \  B ) )  =  B
43difeq2i 3451 . . . 4  |-  ( ( _V  \  A ) 
\  ( _V  \ 
( _V  \  B
) ) )  =  ( ( _V  \  A )  \  B
)
52, 4eqtr2i 2464 . . 3  |-  ( ( _V  \  A ) 
\  B )  =  ( ( _V  \  A )  i^i  ( _V  \  B ) )
65difeq2i 3451 . 2  |-  ( _V 
\  ( ( _V 
\  A )  \  B ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )
71, 6eqtri 2463 1  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1654   _Vcvv 2965    \ cdif 3306    u. cun 3307    i^i cin 3308
This theorem is referenced by:  difundi  3581  unvdif  3730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2717  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316
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