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 ) ∪ ( V ∖ 𝐴 ) )
2		ssv	⊢ 𝐴 ⊆ V
3		ssdif0	⊢ ( 𝐴 ⊆ V ↔ ( 𝐴 ∖ V ) = ∅ )
4	2 3	mpbi	⊢ ( 𝐴 ∖ V ) = ∅
5	4	uneq1i	⊢ ( ( 𝐴 ∖ V ) ∪ ( V ∖ 𝐴 ) ) = ( ∅ ∪ ( V ∖ 𝐴 ) )
6		uncom	⊢ ( ∅ ∪ ( V ∖ 𝐴 ) ) = ( ( V ∖ 𝐴 ) ∪ ∅ )
7		un0	⊢ ( ( V ∖ 𝐴 ) ∪ ∅ ) = ( V ∖ 𝐴 )
8	6 7	eqtri	⊢ ( ∅ ∪ ( V ∖ 𝐴 ) ) = ( V ∖ 𝐴 )
9	5 8	eqtri	⊢ ( ( 𝐴 ∖ V ) ∪ ( V ∖ 𝐴 ) ) = ( V ∖ 𝐴 )
10	1 9	eqtri	⊢ ( 𝐴 △ V ) = ( V ∖ 𝐴 )