Let $S=\{x\in \mathbb{R}^n| p(x)=0, \nabla p(x)\not= 0, p(x)\in C^{\infty}\}$. We defined the distribution $\delta(p(x))\in \mathcal{D}'(\mathbb{R}^n)$ on $S$ as follows: for all $\varphi\in C_{0}^{\infty}(\mathbb{R}^n)$ $$ \langle \delta(p(x)),\varphi\rangle:=\int_{p(x)=0}\varphi(x) dS_{x}.$$ Show that $$ \text{sing}~\text{supp}~\delta(p(x))=\text{supp}~\delta(p(x))=\{x|p(x)=0\}.$$ Here $\text{sing}~\text{supp}~u$ is the singular support of a distribution $u$.
I only know that $ \text{supp}~\delta(p(x))=\{x|p(x)=0\}$ and $\text{sing}~\text{supp}~\delta(p(x))\subset\text{supp}~\delta(p(x))$, but I don't know how to prove that $\text{supp}~\delta(p(x))\subset \text{sing}~\text{supp}~\delta(p(x))$. Can someone help me? Thank you very much in advance!