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

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

Hence the result.

$\blacksquare$


Sources

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