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Theorem undifv 3646
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 3523 . 2  |-  ( A  u.  ( _V  \  A ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  ( _V  \  A
) ) ) )
2 disjdif 3644 . . 3  |-  ( ( _V  \  A )  i^i  ( _V  \ 
( _V  \  A
) ) )  =  (/)
32difeq2i 3406 . 2  |-  ( _V 
\  ( ( _V 
\  A )  i^i  ( _V  \  ( _V  \  A ) ) ) )  =  ( _V  \  (/) )
4 dif0 3642 . 2  |-  ( _V 
\  (/) )  =  _V
51, 3, 43eqtri 2412 1  |-  ( A  u.  ( _V  \  A ) )  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2900    \ cdif 3261    u. cun 3262    i^i cin 3263   (/)c0 3572
This theorem is referenced by:  undif1  3647  dfif4  3694  hashf  11553  fullfunfnv  25510  hfext  25839
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 2369
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 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573
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