# Strictly Monotone Mapping with Totally Ordered Domain is Injective

## Theorem

Let $\struct {S, \preceq_1}$ be a totally ordered set.

Let $\struct {T, \preceq_2}$ be an ordered set.

Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be a strictly monotone mapping.

Then $\phi$ is injective.

## Proof

 $\displaystyle x, y$ $\in$ $\displaystyle S$ $\, \displaystyle \land \,$ $\displaystyle x$ $\ne$ $\displaystyle y$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $\prec_1$ $\displaystyle y$ $\, \displaystyle \lor \,$ $\displaystyle y$ $\prec_1$ $\displaystyle x$ Trichotomy Law $\displaystyle \leadsto \ \$ $\displaystyle \map \phi x$ $\prec_2$ $\displaystyle \map \phi y$ $\, \displaystyle \lor \,$ $\displaystyle \map \phi y$ $\prec_2$ $\displaystyle \map \phi x$ $\phi$ is strictly monotone $\displaystyle \leadsto \ \$ $\displaystyle \map \phi x$ $\ne$ $\displaystyle \map \phi y$ Definition of $\prec_2$

Hence the result.

$\blacksquare$