A Solution to a Question of Bouchard

We prove an intersection lemma for finite indexed families of finite sets, allowing repetitions. If c,m≥1 and m≥\ |i∈[m]Cᵢ|+c, then there is a nonempty set of indices K⊆[m] such that |i∈ KCᵢ|=|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, UCₓ holds for every union-closed family with a nonempty n-element universe and every integer 1≤ x≤ n/2+1.