MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfun3 Unicode version

Theorem dfun3 3566
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 3563 . 2  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  \  B )
)
2 dfin2 3564 . . . 4  |-  ( ( _V  \  A )  i^i  ( _V  \  B ) )  =  ( ( _V  \  A )  \  ( _V  \  ( _V  \  B ) ) )
3 ddif 3466 . . . . 5  |-  ( _V 
\  ( _V  \  B ) )  =  B
43difeq2i 3449 . . . 4  |-  ( ( _V  \  A ) 
\  ( _V  \ 
( _V  \  B
) ) )  =  ( ( _V  \  A )  \  B
)
52, 4eqtr2i 2451 . . 3  |-  ( ( _V  \  A ) 
\  B )  =  ( ( _V  \  A )  i^i  ( _V  \  B ) )
65difeq2i 3449 . 2  |-  ( _V 
\  ( ( _V 
\  A )  \  B ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )
71, 6eqtri 2450 1  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   _Vcvv 2943    \ cdif 3304    u. cun 3305    i^i cin 3306
This theorem is referenced by:  difundi  3580  undifv  3689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ral 2697  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314
  Copyright terms: Public domain W3C validator