I am reading the book Mathematical Logic and Model Theory by Prestel and Delzell and after they talked about first order semantics there was a section on the axiomatization of set theory. They present the foundation axiom in the following way:
$$\forall x (x\neq\emptyset \longrightarrow \exists z (z\in x \wedge z \cap x=\emptyset))$$
Then they go on to say that $z\cap x$ (for intersection) denotes a set whose existence is proved using the axiom of replacement and whose uniqueness is guaranteed by the axiom of extensionality.
I don't get this at all. The more I think about it the more confused I am.They don't define $z\cap x$. Is the axiom supposed to be the definition of it or are we supposed to take the usual definition?
Could someone explain how to prove existence and uniqueness of $z\cap x$ ?