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Theorem undifv 3645
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 3522 . 2  |-  ( A  u.  ( _V  \  A ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  ( _V  \  A
) ) ) )
2 disjdif 3643 . . 3  |-  ( ( _V  \  A )  i^i  ( _V  \ 
( _V  \  A
) ) )  =  (/)
32difeq2i 3405 . 2  |-  ( _V 
\  ( ( _V 
\  A )  i^i  ( _V  \  ( _V  \  A ) ) ) )  =  ( _V  \  (/) )
4 dif0 3641 . 2  |-  ( _V 
\  (/) )  =  _V
51, 3, 43eqtri 2411 1  |-  ( A  u.  ( _V  \  A ) )  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2899    \ cdif 3260    u. cun 3261    i^i cin 3262   (/)c0 3571
This theorem is referenced by:  undif1  3646  dfif4  3693  hashf  11552  fullfunfnv  25509  hfext  25838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572
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