State and prove Sperner's lemma on antichains.

The family $\mathcal{A} \subset \mathcal{P}[n]$ is said to split $[n]$ if, for all distinct $i, j \in[n]$, there exists $A \in \mathcal{A}$ with $i \in A$ but $j \notin A$. Prove that if $\mathcal{A}$ splits $[n]$ then $n \leq\left(\begin{array}{c}a \\ \lfloor a / 2\rfloor\end{array}\right)$, where $a=|\mathcal{A}|$.

Show moreover that, if $\mathcal{A}$ splits $[n]$ and no element of $[n]$ is in more than $k<\lfloor a / 2\rfloor$ members of $\mathcal{A}$, then $n \leq\left(\begin{array}{l}a \\ k\end{array}\right)$.