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Theorem unvdif 3856
Description: The union of a class and its complement is the universe. Theorem 5.1(5) of [Stoll] p. 17. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
unvdif  |-  ( A  u.  ( _V  \  A ) )  =  _V

Proof of Theorem unvdif
StepHypRef Expression
1 dfun3 3691 . 2  |-  ( A  u.  ( _V  \  A ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  ( _V  \  A
) ) ) )
2 disjdif 3854 . . 3  |-  ( ( _V  \  A )  i^i  ( _V  \ 
( _V  \  A
) ) )  =  (/)
32difeq2i 3574 . 2  |-  ( _V 
\  ( ( _V 
\  A )  i^i  ( _V  \  ( _V  \  A ) ) ) )  =  ( _V  \  (/) )
4 dif0 3852 . 2  |-  ( _V 
\  (/) )  =  _V
51, 3, 43eqtri 2485 1  |-  ( A  u.  ( _V  \  A ) )  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   _Vcvv 3072    \ cdif 3428    u. cun 3429    i^i cin 3430   (/)c0 3740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741
This theorem is referenced by:  undif1  3857  dfif4  3907  hashf  12222  fullfunfnv  28116  hfext  28360
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