  A set of sets is called a family. Two families $\mathcal{A}$ and
$\mathcal{B}$ of sets are said to be cross-intersecting if each member of
$\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$
and $k$ with $1 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of
subsets of $[n] = \{1, \dots, n\}$ that have at most $k$ elements. We show that
if $\mathcal{A}$ is a non-empty subfamily of ${[n] \choose \leq r}$,
$\mathcal{B}$ is a non-empty subfamily of ${[n] \choose \leq s}$, $r \leq s$,
and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then
\[|\mathcal{A}| + |\mathcal{B}| \leq 1 + \sum_{i=1}^s \left({n \choose i} -
{n-r \choose i} \right),\] and equality holds if $\mathcal{A} = \{[r]\}$ and
$\mathcal{B}$ is the family of sets in ${[n] \choose \leq s}$ that intersect
$[r]$.
