The concept of d-separation holds a pivotal role in causality theory, serving\nas a fundamental tool for deriving conditional independence properties from\ncausal graphs. Pearl defined the d-separation of two subsets conditionally on a\nthird one. In this study, we present a novel perspective by showing i) how the\nd-separation can be extended beyond acyclic graphs, possibly infinite, and ii)\nhow it can be expressed and characterized as a binary relation between\nvertices. Compared to the typical perspectives in causality theory, our\nequivalence opens the door to more compact and computational proofing\ntechniques, because the language of binary relations is well adapted to\nequational reasoning. Additionally, and of independent interest, the proofs of\nthe results presented in this paper are checked with the Coq proof assistant.\n