Let $X$ be a smooth manifold, let $u \in \mathcal{D}'(X)$, and let $s,t \in \mathbb{R}$ satisfy $s \le t$. For each $r \in \mathbb{R}$, let $WF^r(u) \subset T^*X \setminus \{0\}$ denote the Sobolev wave front set defined by the condition that a covector $q \in T^*X \setminus \{0\}$ is not in $WF^r(u)$ precisely when there exists an elliptic microlocal cutoff $A$ at $q$ such that $Au \in H^r_{\mathrm{loc}}(X)$. Then

\begin{align*} WF^s(u) \subset WF^t(u). \end{align*}