Theorem symdifcom

Description: Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012)

	Ref	Expression
Assertion	symdifcom	⊢ ( 𝐴 △ 𝐵 ) = ( 𝐵 △ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		uncom	⊢ ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐵 ∖ 𝐴 ) ) = ( ( 𝐵 ∖ 𝐴 ) ∪ ( 𝐴 ∖ 𝐵 ) )
2		df-symdif	⊢ ( 𝐴 △ 𝐵 ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐵 ∖ 𝐴 ) )
3		df-symdif	⊢ ( 𝐵 △ 𝐴 ) = ( ( 𝐵 ∖ 𝐴 ) ∪ ( 𝐴 ∖ 𝐵 ) )
4	1 2 3	3eqtr4i	⊢ ( 𝐴 △ 𝐵 ) = ( 𝐵 △ 𝐴 )