Fix $A, B \in \mathbf{\Delta}_{1}^{1}$ (i.e. they're Borel). Is the statement $A \subseteq B$ generally only $\mathbf{\Pi}^{1}_{1}$ (at best)? Of course, it's $\mathbf{\Pi}^{1}_{1}$ via $\forall x[x \in A \rightarrow x \in B]$.
EDIT: Here's my situation in more detail. For each continuous function $f\in C[0,1]$, let $S_f \subseteq [0,1]$ be a $\Sigma_{3}^{0}(f)$ set (that is, a set that is lightface sigma zero three in $f$) and for each $e\in \omega$, the set $Q_{e}\subset \mathbb{Q}$ is finite and computable (uniformly in $e$). I want to know the complexity of $\exists e[S_{f}\subseteq Q_{e}]$. In general, must I have a set (i.e. real number) quantifier to express that subset relation?