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Theorem symdifV 29450
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 29443 . 2  |-  ( A(++)
_V )  =  ( ( A  \  _V )  u.  ( _V  \  A ) )
2 ssv 3509 . . . . 5  |-  A  C_  _V
3 ssdif0 3871 . . . . 5  |-  ( A 
C_  _V  <->  ( A  \  _V )  =  (/) )
42, 3mpbi 208 . . . 4  |-  ( A 
\  _V )  =  (/)
54uneq1i 3639 . . 3  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  (
(/)  u.  ( _V  \  A ) )
6 uncom 3633 . . . 4  |-  ( (/)  u.  ( _V  \  A
) )  =  ( ( _V  \  A
)  u.  (/) )
7 un0 3796 . . . 4  |-  ( ( _V  \  A )  u.  (/) )  =  ( _V  \  A )
86, 7eqtri 2472 . . 3  |-  ( (/)  u.  ( _V  \  A
) )  =  ( _V  \  A )
95, 8eqtri 2472 . 2  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  ( _V  \  A )
101, 9eqtri 2472 1  |-  ( A(++)
_V )  =  ( _V  \  A )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383   _Vcvv 3095    \ cdif 3458    u. cun 3459    C_ wss 3461   (/)c0 3770  (++)csymdif 29442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-symdif 29443
This theorem is referenced by: (None)
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