We want to prove that for each $\displaystyle A,B \in \mathcal{P}(X) $, there exists a unique set $\displaystyle C = A \Delta B $ such that $\displaystyle A \Delta C = B $.

So $\displaystyle A \Delta C = B \Rightarrow (A-C) \cup (C-A) = B \Rightarrow A-C \subseteq B $ and $\displaystyle C-A \subseteq B \Rightarrow $ $\displaystyle C = (A-B) \cup (B-A) \Rightarrow C = A \Delta B $?

Obviously $\displaystyle C = A \Delta B $ because $\displaystyle A \Delta (A \Delta B) = (A \Delta A) \Delta B = \emptyset \Delta B = B $ (by associative law). But that only proves existence and not uniqueness.

Thanks