What is the correct way to represent the intersection of two non-null events?

Question

I would like to define a set for two sets $A$ and $B$ such that:

$$ \{x: \{x \in A\} \neq \emptyset \ \ \cap \ \ \{x \in B\}\neq \emptyset \} $$

That is, I would like find the collection of events $x$ that are simultaneously not the empty set under both $A$ and $B$. Is there a more elegant way to write this?