In axiomatic set theory, we have Axiom of empty set: $\exists \varnothing \forall x ( x\notin \varnothing)$. Is there any equivalent statement without the use of quantifiers? For example, $ \exists x\in y (P (x)) $ is equivalent to $ x\in y\: \And\: P (x) $ and $\forall x\in y (P (x) $ is equivalent to $ x\in y\Rightarrow P (x) $ (or are they?). Any try to clear things up is appreciated.