Let $n > k$ be positive integers and let $F$ be a family of finite sets with the following properties: i. $F$ contains at least $\binom{n}{k}+ 1$ distinct sets containing exactly $k$ elements; ii. For any two sets $A, B \in F$ their union, i.e., $A \cup B$ also belongs to $F$. Prove that $F$ contains at least three sets with at least $n$ elements.