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Theorem symdifv 4347
Description: Symmetric difference with the universal class. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdifv  |-  ( A  /_\  _V )  =  ( _V  \  A )

Proof of Theorem symdifv
StepHypRef Expression
1 df-symdif 3654 . 2  |-  ( A  /_\  _V )  =  ( ( A  \  _V )  u.  ( _V  \  A ) )
2 ssv 3438 . . . . 5  |-  A  C_  _V
3 ssdif0 3741 . . . . 5  |-  ( A 
C_  _V  <->  ( A  \  _V )  =  (/) )
42, 3mpbi 213 . . . 4  |-  ( A 
\  _V )  =  (/)
54uneq1i 3575 . . 3  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  (
(/)  u.  ( _V  \  A ) )
6 uncom 3569 . . . 4  |-  ( (/)  u.  ( _V  \  A
) )  =  ( ( _V  \  A
)  u.  (/) )
7 un0 3762 . . . 4  |-  ( ( _V  \  A )  u.  (/) )  =  ( _V  \  A )
86, 7eqtri 2493 . . 3  |-  ( (/)  u.  ( _V  \  A
) )  =  ( _V  \  A )
95, 8eqtri 2493 . 2  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  ( _V  \  A )
101, 9eqtri 2493 1  |-  ( A  /_\  _V )  =  ( _V  \  A )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   _Vcvv 3031    \ cdif 3387    u. cun 3388    C_ wss 3390    /_\ csymdif 3653   (/)c0 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-symdif 3654  df-nul 3723
This theorem is referenced by: (None)
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