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Theorem symdifV 24440
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 24433 . 2  |-  ( A(++)
_V )  =  ( ( A  \  _V )  u.  ( _V  \  A ) )
2 ssv 3211 . . . . 5  |-  A  C_  _V
3 ssdif0 3526 . . . . 5  |-  ( A 
C_  _V  <->  ( A  \  _V )  =  (/) )
42, 3mpbi 199 . . . 4  |-  ( A 
\  _V )  =  (/)
54uneq1i 3338 . . 3  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  (
(/)  u.  ( _V  \  A ) )
6 uncom 3332 . . . 4  |-  ( (/)  u.  ( _V  \  A
) )  =  ( ( _V  \  A
)  u.  (/) )
7 un0 3492 . . . 4  |-  ( ( _V  \  A )  u.  (/) )  =  ( _V  \  A )
86, 7eqtri 2316 . . 3  |-  ( (/)  u.  ( _V  \  A
) )  =  ( _V  \  A )
95, 8eqtri 2316 . 2  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  ( _V  \  A )
101, 9eqtri 2316 1  |-  ( A(++)
_V )  =  ( _V  \  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468  (++)csymdif 24432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-symdif 24433
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