 Separating family of disjoint set pairs

A family $(A_i,B_i{)}_{i=1}^m$ of pairs of disjoint subsets of a finite set separates its points if each two distinct points lie on opposite sides of at least one pair. Empty sides are allowed by this definition. Let $d(x)$ count pairs containing $x$. Associate to $x$ the subcube of $\{0,1{\}}^m$ fixing coordinate $i$ to zero on $A_i$ and one on $B_i$. Separation makes these subcubes disjoint and yields the disjoint subcube packing inequality.