I came across this question:
Show that $A \cap \emptyset= \emptyset$
The solution is that since (earlier in the book I am following) $A\cap B\subseteq B$, this proves that $A\cap \emptyset\subseteq\emptyset$.
Also that the empty set is a member of all sets, so that proves $\emptyset\subseteq A\cap\emptyset$, by the axiom of extensionality, this proves that $A\cap\emptyset=\emptyset$.
But I found this weird: for all $x, A\cap B$ iff $x\in A$ and $x\in B$. So surely $A\cap\emptyset\subseteq\emptyset$ means that $x\in\emptyset$, which contradicts with the fact that there is no element in the empty set!
Could anyone help please?