Theorem dfsymdif3

Description: Alternate definition of the symmetric difference, given in Example 4.1 of Stoll p. 262 (the original definition corresponds to Stoll p. 13). (Contributed by NM, 17-Aug-2004) (Revised by BJ, 30-Apr-2020)

	Ref	Expression
Assertion	dfsymdif3	|- ( A /_\ B ) = ( ( A u. B ) \ ( A i^i B ) )

Proof

Step	Hyp	Ref	Expression
1		difin	|- ( A \ ( A i^i B ) ) = ( A \ B )
2		incom	|- ( A i^i B ) = ( B i^i A )
3	2	difeq2i	|- ( B \ ( A i^i B ) ) = ( B \ ( B i^i A ) )
4		difin	|- ( B \ ( B i^i A ) ) = ( B \ A )
5	3 4	eqtri	|- ( B \ ( A i^i B ) ) = ( B \ A )
6	1 5	uneq12i	|- ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) ) = ( ( A \ B ) u. ( B \ A ) )
7		difundir	|- ( ( A u. B ) \ ( A i^i B ) ) = ( ( A \ ( A i^i B ) ) u. ( B \ ( A i^i B ) ) )
8		df-symdif	|- ( A /_\ B ) = ( ( A \ B ) u. ( B \ A ) )
9	6 7 8	3eqtr4ri	|- ( A /_\ B ) = ( ( A u. B ) \ ( A i^i B ) )