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Theorem dfun3 3717
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 3714 . 2  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  \  B )
)
2 dfin2 3715 . . . 4  |-  ( ( _V  \  A )  i^i  ( _V  \  B ) )  =  ( ( _V  \  A )  \  ( _V  \  ( _V  \  B ) ) )
3 ddif 3603 . . . . 5  |-  ( _V 
\  ( _V  \  B ) )  =  B
43difeq2i 3586 . . . 4  |-  ( ( _V  \  A ) 
\  ( _V  \ 
( _V  \  B
) ) )  =  ( ( _V  \  A )  \  B
)
52, 4eqtr2i 2459 . . 3  |-  ( ( _V  \  A ) 
\  B )  =  ( ( _V  \  A )  i^i  ( _V  \  B ) )
65difeq2i 3586 . 2  |-  ( _V 
\  ( ( _V 
\  A )  \  B ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )
71, 6eqtri 2458 1  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   _Vcvv 3087    \ cdif 3439    u. cun 3440    i^i cin 3441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449
This theorem is referenced by:  difundi  3731  unvdif  3875
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