Consider a maximal sperner family $F$ of subsets of $X = \{ 1,2,3 \ldots n \}$. I need to prove that this family intersects with each chain of subsets exactly once.
Each chain is defined as : $$\phi= A_0 \subset A_1 \ldots \subset A_n = X $$ where $A_i's$ are subsets of $X$. The chain can not intersect at two points with $F$ from the definition of Sperner families. But how to that F intersects once with every chain. Total number of chains are $n!$.