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Theorem unvdif 3866
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 3708 . 2  |-  ( A  u.  ( _V  \  A ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  ( _V  \  A
) ) ) )
2 disjdif 3864 . . 3  |-  ( ( _V  \  A )  i^i  ( _V  \ 
( _V  \  A
) ) )  =  (/)
32difeq2i 3577 . 2  |-  ( _V 
\  ( ( _V 
\  A )  i^i  ( _V  \  ( _V  \  A ) ) ) )  =  ( _V  \  (/) )
4 dif0 3862 . 2  |-  ( _V 
\  (/) )  =  _V
51, 3, 43eqtri 2453 1  |-  ( A  u.  ( _V  \  A ) )  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   _Vcvv 3078    \ cdif 3430    u. cun 3431    i^i cin 3432   (/)c0 3758
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 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
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 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759
This theorem is referenced by:  undif1  3867  dfif4  3921  hashf  12515  fullfunfnv  30699  hfext  30936
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