Union bound for intersecting families ($|{\bigcup }_{i=1}^k{\mathcal{A}}_i|\le 2^n-2^{n-k}$)

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 $k\le n$, the families of sets containing coordinate $i$, for $1\le i\le k$, attain equality.