I'm looking for a proof critique for the following problem from Hrbacek & Jech's "Introduction to Set Theory". It seems a bit cumbersome, but perhaps that's just the nature of all things axiomatic.
Let A and B be sets. Prove that there exists a unique set, $C$, such that $x\in C$ if and only if either $x\in A$ and $x\notin B$, or $x\in B$ and $x\notin A$.
To show existence: $A,B$ are, by hypothesis, sets. Let $\mathbf P(x,B)$ be the property of $x$, with parameter $B$, given by "$x\notin B$." Then by the comprehension schema axiom, there exists some set, $A_0$, such that $x\in A_0$ if and only if $x\in A$ and $\mathbf P(x,B)$. Let $\mathbf Q(x,A)$ be the property of $x$, with parameter $A$, given by "$x\notin A$." Analogously, then, there exists some set, $B_0$, such that $x\in B_0$ if and only if $x\in B$ and $\mathbf Q(x,A)$. Now, applying the axiom of pair to $A_0,B_0$, results in the existence of some set, $C_0$, such that $x\in C_0$ if and only if $x=A_0$ or $x=B_0$; more simply, $C_0 = \{A_0,B_0\}$. Finally, applying the axiom of union to $C_0$, we may conclude the existence of the desired set, $C$, where $x\in C$ if and only if $x\in S$ for some $S\in C_0$. In other words, $x\in C$ if and only if $x\in A_0$ or $x\in B_0$, since $C_0 = \{A_0,B_0\}.$ Substituting the meanings of "$x\in A_0$" and "$x\in B_0$" from above, we reach, finally: $x\in C$ if and only if $x\in A$ and $x\notin B$ or $x\in B$ and $x\notin A$.
To show uniqueness: Let $C^*$ be any other set such that $x\in C^*$ if and only if either $x\in A$ and $x\notin B$ or $x\in B$ and $x\notin A$. Then, since also $x\in C$ if and only if either $x\in A$ and $x\notin B$ or $x\in B$ and $x\notin A$, it follows that $x\in C^*$ if and only if $x\in C$. Hence, $C=C^*$ by the axiom of extensionality.