Strictly Monotone Mapping with Totally Ordered Domain is Injective

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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.

$\blacksquare$


Sources