Let $\mathcal{F}$ be a Sperner family, that is, the collection of subsets of $\{1,2,\dotsc,n\}$ such that no set contains any other subset. Then \begin{equation*} \sum_{X\in\mathcal{F}} \frac{1}{\binom{n}{\abs{X}}}\leq 1. \end{equation*}This inequality is known as LYM inequality by the names of three people that independently discovered it: Lubell[2], Yamamoto[4], Meshalkin[3].

Since $\binom{n}{k}\leq \binom{n}{\floor{n/2}}$ for every integer $k$ LYM inequality tells us that $\abs{F}/\binom{n}{\floor{n/2}}\leq 1$ which is Sperner's theorem.

References

1

Konrad Engel.
Sperner theory, volume 65 of Encyclopedia of Mathematics and Its Applications.
Cambridge University Press.
Zbl 0868.05001.

2

David Lubell.
A short proof of Sperner's lemma.
J. Comb. Theory, 1:299, 1966.
Zbl 0151.01503.

3

Lev D. Meshalkin.
Generalization of Sperner's theorem on the number of subsets of a finite set.
Teor. Veroyatn. Primen., 8:219-220, 1963.
Zbl 0123.36303.

4

Koichi Yamamoto.
Logarithmic order of free distributive lattice.
J. Math. Soc. Japan, 6:343-353, 1954.
Zbl 0056.26301.