Let $X$ be a smooth $n$-dimensional manifold, let $u \in \mathcal{D}'(X)$ be a distribution, and let $(x_0,\xi_0) \in T^*X \setminus 0$. Assume $WF(u)$ is defined by the local conic rapid-decay condition for Fourier transforms of compactly supported coordinate localizations. Assume $\Psi^0(X)$ denotes the class of properly supported order-zero pseudodifferential operators on $X$, with homogeneous principal symbol defined on $T^*X \setminus 0$, and assume the standard pseudodifferential calculus on $X$: coordinate invariance of principal symbols and wave front sets, conic symbol cutoffs, smoothing remainders, and the microlocal elliptic parametrix theorem. An operator $A \in \Psi^0(X)$ is elliptic at $(x_0,\xi_0)$ if its principal symbol is nonzero at $(x_0,\xi_0)$. Then

\begin{align*} (x_0,\xi_0) \notin WF(u) \end{align*}

if and only if there exists an operator $A \in \Psi^0(X)$ elliptic at $(x_0,\xi_0)$ such that

\begin{align*} Au \in C^\infty(X). \end{align*}