Find set configurations that imply FUNC

Introduction

One of the first observations on the conjecture is that if a union closed family contains a singleton set {x}, then the set [math]\displaystyle{ \mathcal{A}_x = \{A \in \mathcal{A} : x \in A\} }[/math], contains at least as many sets as [math]\displaystyle{ \mathcal{A}_x = \{A \in \mathcal{A} : x \not in A\} }[/math]