 Families compressed in every binary section

If both sections for every coordinate are binary initial segments, then the family itself is a binary initial segment, apart from the possible family $(\mathcal{P}([n-1])\setminus \{[n-1]\})\cup \{\{n\}\}$. Any missing vertex earlier than a present vertex must disagree with it in every coordinate; the two are therefore complementary and consecutive in binary order on the discrete cube, forcing this sole exceptional shape.