Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  symdifV Structured version   Unicode version

Theorem symdifV 25662
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 25655 . 2  |-  ( A(++)
_V )  =  ( ( A  \  _V )  u.  ( _V  \  A ) )
2 ssv 3360 . . . . 5  |-  A  C_  _V
3 ssdif0 3678 . . . . 5  |-  ( A 
C_  _V  <->  ( A  \  _V )  =  (/) )
42, 3mpbi 200 . . . 4  |-  ( A 
\  _V )  =  (/)
54uneq1i 3489 . . 3  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  (
(/)  u.  ( _V  \  A ) )
6 uncom 3483 . . . 4  |-  ( (/)  u.  ( _V  \  A
) )  =  ( ( _V  \  A
)  u.  (/) )
7 un0 3644 . . . 4  |-  ( ( _V  \  A )  u.  (/) )  =  ( _V  \  A )
86, 7eqtri 2455 . . 3  |-  ( (/)  u.  ( _V  \  A
) )  =  ( _V  \  A )
95, 8eqtri 2455 . 2  |-  ( ( A  \  _V )  u.  ( _V  \  A
) )  =  ( _V  \  A )
101, 9eqtri 2455 1  |-  ( A(++)
_V )  =  ( _V  \  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   (/)c0 3620  (++)csymdif 25654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-symdif 25655
  Copyright terms: Public domain W3C validator