It is known that adjunctive set theory interprets Robinson arithmetic, and that extensionality is not needed for that. (Montagna and Mancini, "A minimal predicative set theory", Notre Dame Journal of Formal Logic, 1994).

I wonder whether it is known if the following weak set theory (without extensionality) also interprets Robinson arithmetic. Axioms:

$\exists y\forall x(x\notin y)$

For every n, $\forall x_1\ldots\forall x_n\exists y\forall z(z\in y\leftrightarrow z=x_1 \vee \ldots\vee z=x_n)$