Theorem symdifv

Description: The symmetric difference with the universal class is the complement. (Contributed by Scott Fenton, 24-Apr-2012)

	Ref	Expression
Assertion	symdifv	|- ( A /_\ _V ) = ( _V \ A )

Proof

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