Trichotomy Law for Real Numbers

Theorem

The real numbers obey the Trichotomy Law.

That is, $\forall a, b \in \R$, exactly one of the following holds:

 $(1)$ $:$ $a$ is greater than $b$: $\displaystyle a > b$ $(2)$ $:$ $a$ is equal to $b$: $\displaystyle a = b$ $(3)$ $:$ $a$ is less than $b$: $\displaystyle a < b$

Proof 1

This follows directly Real Numbers form Totally Ordered Field.

$\blacksquare$

Proof 2

$\le$ is a total ordering on $\R$.

The trichotomy follows directly from Trichotomy Law.

$\blacksquare$

Also known as

The Trichotomy Law can also be seen referred to as the trichotomy principle.