I am studying Elements of Set Theory by Enderton and don't understand how the existential quantifier in the Empty Set Axiom guarantees its existence.
The axiom reads
$$\exists B(\forall x \neg(x\in B)).$$
From my logic book, I understand this quantifier to represent the disjunction of the wff (here, $\forall x \neg(x\in B))$) over the domain of quantification, i.e.
$$\bigvee B(\forall x\neg(x\in B)).$$
If the domain is void, then there are no disjuncts for the above axiom, and I would assume it's truth value is undefined, or that it is not even a sentence in the formal language.
To restate more colloquially, how are we bootstrapping a void domain with an object that by definition comes into existence by the use of a quantifier that must range over this void domain?