= Solution
In the <four functions theorem>, take $\alpha=\mathbf1_{\mathcal A}$, $\beta=\mathbf1_{\mathcal B}$, $\gamma=\mathbf1_{\mathcal A\vee\mathcal B}$ and $\delta=\mathbf1_{\mathcal A\wedge\mathcal B}$, where the <union and intersection of set families> are the collections of all pairwise unions and intersections.
If $\alpha(A)\beta(B)=1$, then $A\cup B\in\mathcal A\vee\mathcal B$ and $A\cap B\in\mathcal A\wedge\mathcal B$, so the pointwise hypothesis holds. Otherwise its left side is zero. Summing these <indicator functions> gives
$$
\boxed{|\mathcal A\vee\mathcal B|\,|\mathcal A\wedge\mathcal B|\ge|\mathcal A|\,|\mathcal B|}.
$$
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