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Theorem dfun3 3409
Description: Union defined in terms of intersection (DeMorgan'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 3406 . 2  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  \  B )
)
2 dfin2 3407 . . . 4  |-  ( ( _V  \  A )  i^i  ( _V  \  B ) )  =  ( ( _V  \  A )  \  ( _V  \  ( _V  \  B ) ) )
3 ddif 3310 . . . . 5  |-  ( _V 
\  ( _V  \  B ) )  =  B
43difeq2i 3293 . . . 4  |-  ( ( _V  \  A ) 
\  ( _V  \ 
( _V  \  B
) ) )  =  ( ( _V  \  A )  \  B
)
52, 4eqtr2i 2306 . . 3  |-  ( ( _V  \  A ) 
\  B )  =  ( ( _V  \  A )  i^i  ( _V  \  B ) )
65difeq2i 3293 . 2  |-  ( _V 
\  ( ( _V 
\  A )  \  B ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )
71, 6eqtri 2305 1  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1624   _Vcvv 2790    \ cdif 3151    u. cun 3152    i^i cin 3153
This theorem is referenced by:  difundi  3423  undifv  3530
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161
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