Is the symmetric difference associative? I'd like to be able to give a proof for it.

I.e. is the following true? $\displaystyle A \oplus (B \oplus C) = (A \oplus B) \oplus C$, where A, B, and C are sets.

To me it seems this is the case, i.e. that the symmetric difference is associative.

Let $\displaystyle a \in (A \oplus (B \oplus C))$ \ $\displaystyle ((A \oplus B) \oplus C)$. I.e. assume that there is an element a in $\displaystyle (A \oplus (B \oplus C))$ that is not in $\displaystyle ((A \oplus B) \oplus C).$

This implies that a must be in only one of A, B or C for this to be true.

But if a is in either A, B or C, but not in two or more; then by definition of the symmetric difference, it must also be in $\displaystyle ((A \oplus B) \oplus C).$ Which is a contradiction -- a cannot be in $\displaystyle (A \oplus (B \oplus C))$ without also being in $\displaystyle ((A \oplus B) \oplus C).$

But how can I give a proof for this?