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Theorem unvdif 3901
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 3736 . 2  |-  ( A  u.  ( _V  \  A ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  ( _V  \  A
) ) ) )
2 disjdif 3899 . . 3  |-  ( ( _V  \  A )  i^i  ( _V  \ 
( _V  \  A
) ) )  =  (/)
32difeq2i 3619 . 2  |-  ( _V 
\  ( ( _V 
\  A )  i^i  ( _V  \  ( _V  \  A ) ) ) )  =  ( _V  \  (/) )
4 dif0 3897 . 2  |-  ( _V 
\  (/) )  =  _V
51, 3, 43eqtri 2500 1  |-  ( A  u.  ( _V  \  A ) )  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786
This theorem is referenced by:  undif1  3902  dfif4  3954  hashf  12376  fullfunfnv  29173  hfext  29417
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