If $(R, {\mathfrak m})$ is a reduced local Noetherian ring with $\dim R = d$, why is the dimension of singular locus of $R$ is strictly less than $d$?
I know that $$ \text{dim Sing} R = \text{sup} \{\text{dim} ~ R/\mathfrak{p} ~~| \mathfrak{p} \in \text{Sing} R \} $$ and $$ \text{Sing} R =\{\mathfrak{p} \in \text{Spec R} ~~ |~~ R_{\mathfrak{p}} ~\text{is not a regular local ring}\}; $$ and that a typical ring is reduced iff $ \cap_{\mathfrak{p} \in Spec R} \mathfrak{p} = \{0\} $, but can not reach any relationship between these informations.
Any help would be thanked.