Suppose we use the following version of the axiom of infinity in ZFC:
$$\exists x:(\varnothing\in x)\wedge(\forall y\in x:\exists z\in x:y\in z).$$
In words, there is set containing the empty set, in which each member is contained in another member. Is this strong enough to prove the existence of a limit ordinal, the smallest of which can then be defined as $\omega?$