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.

In constructive mathematics, if $A$ is equipped with a tight apartness $\#$, we say that $\sim$ is strongly irrelexive if only distinct elements of $A$ are related:

$$\forall (x:A),\forall (y:A),x\sim y\Rightarrow x\#y.$$\forall (x: A),\; \forall (y: A),\; x \sim y \;\Rightarrow x \# y .

Since $\#$ is irrelexive itself, any strongly irrelexive relation must be irrelexive. The converse holds using excluded middle, through which every set has a unique tight apartness.