nLab
irreflexive relation

A (binary) relation $\sim$ on a set $A$ is irreflexive if no element of $A$ is related to itself:

$$\forall (x:A),x\nsim x$$\forall (x: A),\; x \nsim x

In the language of the $2$-poset Rel of sets and relations, a relation $R:A\to A$ is irreflexive if it is disjoint from the identity relation on $A$:

$${id}_A\cap R\subseteq \varnothing$$\id_A \cap R \subseteq \empty

Of course, this containment is in fact an equality.