= Union bound for intersecting families
{title2=$|\bigcup_{i=1}^k\mathcal A_i|\leq2^n-2^{n-k}$}
For <intersecting families> of subsets of $[n]$, take each <upward closure of a set family>. It remains intersecting and contains at most one set in each complementary pair, so its complement as a family has density at least one half. Those complements are decreasing <set families>. Repeated <Harris-Kleitman inequality> gives intersection density at least $2^{-k}$, proving the displayed bound. For $k\leq n$, the families of sets containing coordinate $i$, for $1\leq i\leq k$, attain equality.
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