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Theorem unvdif 3845
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 3687 . 2  |-  ( A  u.  ( _V  \  A ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  ( _V  \  A
) ) ) )
2 disjdif 3843 . . 3  |-  ( ( _V  \  A )  i^i  ( _V  \ 
( _V  \  A
) ) )  =  (/)
32difeq2i 3557 . 2  |-  ( _V 
\  ( ( _V 
\  A )  i^i  ( _V  \  ( _V  \  A ) ) ) )  =  ( _V  \  (/) )
4 dif0 3841 . 2  |-  ( _V 
\  (/) )  =  _V
51, 3, 43eqtri 2435 1  |-  ( A  u.  ( _V  \  A ) )  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   _Vcvv 3058    \ cdif 3410    u. cun 3411    i^i cin 3412   (/)c0 3737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738
This theorem is referenced by:  undif1  3846  dfif4  3899  hashf  12366  fullfunfnv  30257  hfext  30509
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