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Theorem symdifV 24369
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 24362 . 2  |-  ( A(++)
_V )  =  ( ( A  \  _V )  u.  ( _V  \  A ) )
2 ssv 3198 . . . . 5  |-  A  C_  _V
3 ssdif0 3513 . . . . 5  |-  ( A 
C_  _V  <->  ( A  \  _V )  =  (/) )
42, 3mpbi 199 . . . 4  |-  ( A 
\  _V )  =  (/)
54uneq1i 3325 . . 3  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  (
(/)  u.  ( _V  \  A ) )
6 uncom 3319 . . . 4  |-  ( (/)  u.  ( _V  \  A
) )  =  ( ( _V  \  A
)  u.  (/) )
7 un0 3479 . . . 4  |-  ( ( _V  \  A )  u.  (/) )  =  ( _V  \  A )
86, 7eqtri 2303 . . 3  |-  ( (/)  u.  ( _V  \  A
) )  =  ( _V  \  A )
95, 8eqtri 2303 . 2  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  ( _V  \  A )
101, 9eqtri 2303 1  |-  ( A(++)
_V )  =  ( _V  \  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455  (++)csymdif 24361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-symdif 24362
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