Consider a countable transitive model $\mathfrak{M}$ of set theory.

 Let $X$ be a definable collection of sets of reals.

 My question is: is the type of nondefinable elements in $X$ is definable over $Th(\mathfrak{M})$ or not.

 (I assume that $X$ is infinite.)
 
PS: note that the type of nondefinable elements of $X$ is the type $p(x)$ containing all statements of the form:
 $$ \varphi(x) \rightarrow \exists y \neq x \; \varphi(y),$$
 for every formula $\varphi(x)$, plus the formula defining $X$.