I was messing around a little bit and I got his claim:
Proposition: Let $X \subseteq A_n$, be an affine variety. Then $\dim X=\text{height }I(X)$, where $I(X)$ is the ideal generated by $X$.
I know this is very similar to the result that if $B$ is an integral domain and a finitely generated $k$-algebra, where $k$ is a field, and $p$ and prime ideal in $k[x_1,...,x_n]$, we have;
$$\text{height }(p)+\dim(B/p)=\dim B$$