Theorem symdifv 4598

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

Step	Hyp	Ref	Expression
1		df-symdif 3844	. 2 |- ( A /_\ _V ) = ( ( A \ _V ) u. ( _V \ A ) )
2		ssv 3625	. . . . 5 |- A C_ _V
3		ssdif0 3942	. . . . 5 |- ( A C_ _V <-> ( A \ _V ) = (/) )
4	2, 3	mpbi 220	. . . 4 |- ( A \ _V ) = (/)
5	4	uneq1i 3763	. . 3 |- ( ( A \ _V ) u. ( _V \ A ) ) = ( (/) u. ( _V \ A ) )
6		uncom 3757	. . . 4 |- ( (/) u. ( _V \ A ) ) = ( ( _V \ A ) u. (/) )
7		un0 3967	. . . 4 |- ( ( _V \ A ) u. (/) ) = ( _V \ A )
8	6, 7	eqtri 2644	. . 3 |- ( (/) u. ( _V \ A ) ) = ( _V \ A )
9	5, 8	eqtri 2644	. 2 |- ( ( A \ _V ) u. ( _V \ A ) ) = ( _V \ A )
10	1, 9	eqtri 2644	1 |- ( A /_\ _V ) = ( _V \ A )

Syntax hints:   = wceq 1483   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   /_\ csymdif 3843   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916

This theorem is referenced by: (None)