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 $X$.