axiom of regularity and empty set

Question

So we have axiom of regularity which says that any set (let's call it $A$) has a subset $B$ such that $A\cap B = \emptyset$. But thinking about such sets as $C = \{1, 2\}$ and $D = \{3, 4\}$ we still have that $C\cap D = \emptyset$. So can it be that $\emptyset$ actually sometimes represents $2$ sets' common element (set) and sometimes represents that these $2$ sets are just disjoint? Or everything is ok with this symbolism?

Answer 1

$\emptyset$ is not a member of $A\cap B$. It is $A\cap B$. So $\emptyset$ represents the set of common elements of $A$ and $B$ but is not a common element itself. Also you got the axiom wrong as explained in comments.