In ZF we define the intersection of two sets $x$ and $y$ as the set $x\cap y:= \{z\in x\,|\,z\in y\}$. If we have a finite nonempty set $x = \{x_1,\,x_2,\dots x_n\}$ then we can define the intersection of $x$ as the set \begin{equation*} \bigcap x := \{z\in x_1\,|\, \forall y\in x: z\in y\}. \end{equation*} But what do we do if $x$ is infinite? We cant simply do $\{z\,|\,\forall y\in x: z\in y\}$, because the axiom schema of specification doesn't allow it. My next intuition would be to say: "Ok, we do the same thing as with finite sets, just replace $x_1$ with an arbitrary element of $x$." But can we really do that? Does it mean using the axiom of choice? And if it does, is it a valid definition?