I'm looking for a citable reference for the following identity involving the Stirling numbers of the second kind $S(n, k)$ stated in [Equation (27)](https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html): For $n \geq 2$,
$$
\sum_{m=1}^n S(n, m) (-1)^m (m-1)! = 0.
$$
Thank you.