A Solution to a Question of Bouchard

We prove an intersection lemma for finite indexed families of finite sets, allowing repetitions. If $c,m\ge1$ and $m\ge\ |\bigcup_{i\in[m]}C_i|+c$, then there is a nonempty set of indices $K\subseteq[m]$ such that $|\bigcap_{i\in K}C_i|=|K|-c$. By taking complements in a suitably enlarged universe, we obtain an affirmative answer to Question 2.2 in Bouchard's paper "Conjectures on union-closed families of sets" (2024). Consequently, $UC_x$ holds for every union-closed family with a nonempty $n$-element universe and every integer $1\le x\le\lfloor n/2\rfloor+1$.