Short proof of two cases of Chv\'atal's conjecture.

Chv\'atal (1974) conjectured that no intersecting family $\mathcal{F}$ in a downset can be larger than the largest star. Kleitman and Magnanti (1974) proved it when $\mathcal{F}$ is contained in the union of two stars, and Czabarka, Hurlbert and Kamat (2017) when $rank(\mathcal{F})\le 3$. We give short self-contained proofs of these two statements.