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Theorem undifv 3503
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 3382 . 2  |-  ( A  u.  ( _V  \  A ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  ( _V  \  A
) ) ) )
2 disjdif 3501 . . 3  |-  ( ( _V  \  A )  i^i  ( _V  \ 
( _V  \  A
) ) )  =  (/)
32difeq2i 3266 . 2  |-  ( _V 
\  ( ( _V 
\  A )  i^i  ( _V  \  ( _V  \  A ) ) ) )  =  ( _V  \  (/) )
4 dif0 3499 . 2  |-  ( _V 
\  (/) )  =  _V
51, 3, 43eqtri 2282 1  |-  ( A  u.  ( _V  \  A ) )  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1619   _Vcvv 2763    \ cdif 3124    u. cun 3125    i^i cin 3126   (/)c0 3430
This theorem is referenced by:  undif1  3504  dfif4  3550  hashf  11311  fullfunfnv  23860  hfext  24189
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431
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