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Theorem symdifV 25387
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 25380 . 2  |-  ( A(++)
_V )  =  ( ( A  \  _V )  u.  ( _V  \  A ) )
2 ssv 3305 . . . . 5  |-  A  C_  _V
3 ssdif0 3623 . . . . 5  |-  ( A 
C_  _V  <->  ( A  \  _V )  =  (/) )
42, 3mpbi 200 . . . 4  |-  ( A 
\  _V )  =  (/)
54uneq1i 3434 . . 3  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  (
(/)  u.  ( _V  \  A ) )
6 uncom 3428 . . . 4  |-  ( (/)  u.  ( _V  \  A
) )  =  ( ( _V  \  A
)  u.  (/) )
7 un0 3589 . . . 4  |-  ( ( _V  \  A )  u.  (/) )  =  ( _V  \  A )
86, 7eqtri 2401 . . 3  |-  ( (/)  u.  ( _V  \  A
) )  =  ( _V  \  A )
95, 8eqtri 2401 . 2  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  ( _V  \  A )
101, 9eqtri 2401 1  |-  ( A(++)
_V )  =  ( _V  \  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2893    \ cdif 3254    u. cun 3255    C_ wss 3257   (/)c0 3565  (++)csymdif 25379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-symdif 25380
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