 Union and intersection of set families

For set families, $\mathcal{A}\vee \mathcal{B}=\{A\cup B:A\in \mathcal{A},B\in \mathcal{B}\}$ and $\mathcal{A}\wedge \mathcal{B}=\{A\cap B:A\in \mathcal{A},B\in \mathcal{B}\}$. These are families of distinct sets, rather than multisets indexed by pairs. The four functions theorem gives $|\mathcal{A}\vee \mathcal{B}||\mathcal{A}\wedge \mathcal{B}|\ge |\mathcal{A}||\mathcal{B}|$. Complementing one input family yields the analogous product bound for pairwise set differences.