Law of trichotomy

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero. More generally, trichotomy is the property of an order relation ${\displaystyle <}$ on a set X that for any x and y, exactly one of the following holds: ${\displaystyle x\ <\ y,\ x\ =\ y}$, or ${\displaystyle x\ >\ y}$.

In mathematical notation, this is

Assuming that the ordering is irreflexive and transitive, this can be simplified to

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