The Naive Set Theory ($\mathsf{NST}$) is an inconsistent, trivial theory, but its positive fragment was shown to be consistent and is basis of e.g. positive set theory $\mathsf{GPK}_{\infty}^{+}$ of Olivier Esser.
In the positive fragment, instances of unrestricted comprehension are restricted to only positive formulas (the smallest class of formulas containing atomic membership and equality formulas and closed under $\land$, $\lor$, $\exists$ and $\forall$):
positive comprehension: $\exists y\forall x(x\in y \leftrightarrow \varphi(x))$ s.t. $y$ is not free in $\varphi$ and $\varphi$ is a positive formula.
My question is the following: Is the sentence $\exists y[\forall x(x\in y\leftrightarrow x\in x)\land y\in y]$ provable in the positive fragment of $\mathsf{NST}$ (or at least in $\mathsf{GPK}_{\infty}^{+}$)?
Just for comparison: the sentence above is not provable in $\mathsf{ZFC}$ (since $\{x:x\in x\}$ is equal to $\emptyset$) nor $\mathsf{NF}$ (since $\{x:x\in x\}$ does not exist). In $\mathsf{ZFC{-}FA}$ it is independent.
