MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  undifv Structured version   Unicode version

Theorem undifv 3694
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 3571 . 2  |-  ( A  u.  ( _V  \  A ) )  =  ( _V  \  (
( _V  \  A
)  i^i  ( _V  \  ( _V  \  A
) ) ) )
2 disjdif 3692 . . 3  |-  ( ( _V  \  A )  i^i  ( _V  \ 
( _V  \  A
) ) )  =  (/)
32difeq2i 3454 . 2  |-  ( _V 
\  ( ( _V 
\  A )  i^i  ( _V  \  ( _V  \  A ) ) ) )  =  ( _V  \  (/) )
4 dif0 3690 . 2  |-  ( _V 
\  (/) )  =  _V
51, 3, 43eqtri 2459 1  |-  ( A  u.  ( _V  \  A ) )  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1652   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311   (/)c0 3620
This theorem is referenced by:  undif1  3695  dfif4  3742  hashf  11617  fullfunfnv  25783  hfext  26116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621
  Copyright terms: Public domain W3C validator