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Theorem undifv 3528
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
undifv  |-  ( A  u.  ( _V  \  A ) )  =  _V

Proof of Theorem undifv
StepHypRef Expression
1 dfun3 3407 . 2  |-  ( A  u.  ( _V  \  A ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  ( _V  \  A
) ) ) )
2 disjdif 3526 . . 3  |-  ( ( _V  \  A )  i^i  ( _V  \ 
( _V  \  A
) ) )  =  (/)
32difeq2i 3291 . 2  |-  ( _V 
\  ( ( _V 
\  A )  i^i  ( _V  \  ( _V  \  A ) ) ) )  =  ( _V  \  (/) )
4 dif0 3524 . 2  |-  ( _V 
\  (/) )  =  _V
51, 3, 43eqtri 2307 1  |-  ( A  u.  ( _V  \  A ) )  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151   (/)c0 3455
This theorem is referenced by:  undif1  3529  dfif4  3576  hashf  11344  fullfunfnv  24484  hfext  24813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456
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