In set theory, the axiom of regularity can be used to prove
$$ \tag{*}\label{*} \forall x:x\notin x. $$
Can the direction be reversed, i.e., can $(\ref{*})$ prove regularity in ZF$\setminus$R?
By ZF$\setminus$R I mean the standard axioms of ZF, without the axiom of regularity. Stated differently, as in the title, is $(\ref{*})$ equivalent to the axiom of regularity in ZF$\setminus$R?