lean_workbook_plus_16747

Let $n\geq 1$ be a positive integer, and $(X_{i})_{i\in [[1,n+1]]}$ a family of non-empty sets of $[[1,n]]$. Does there exist two subsets $I$ and $J$ such that $I\cap J$ is empty of $[[1,n+1]]$ such that: $\cup_{i \in I} X_{i} = \cup_{j \in J} X_{j}$?