Erdős-Ko-Rado theorem from shadows

For an intersecting family of $r$-sets with $n\ge 2r$, the rank-$r$ lower shadow of its complement family is disjoint from the original family. Iterating the Kruskal-Katona theorem shows that exceeding $\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$ original members would force this shadow to have more than $\left(\genfrac{}{}{0ex}{}{n-1}{r}\right)$ members, contradicting the total number $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ of $r$-sets.