Consider a ctm $\mathfrak{M}$ of set theory.

Let $X$ be a projectively definable collection of sets of reals (please see my earlier question "Projectively definable family of sets of reals" for the precise definition of projectively definable family 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 $X$.