 Markov blanket D-separation

In a Directed acyclic graph, let $B$ contain the parents, children, and other parents of children of vertex $k$. Then $k$ is D-separated from all remaining vertices by $B$. Every path out of $k$ meets a conditioned noncollider either at a parent, at a child, or at another parent just after a conditioned child-collider. The global Markov property for a directed acyclic graph gives $Z_k\perp Z_{(B\cup \{k\}{)}^c}\mid Z_B$.