Strictly Order-Preserving and Order-Reversing Mapping on Strictly Totally Ordered Set is Injection

Theorem
Let $\struct {S, \prec_1}$ and $\struct {T, \prec_2}$ be strictly totally ordered sets.

Let $\phi: S \to T$ be a mapping.

Let $\pi: S \to T$ be a mapping with the property that:
 * $\forall x, y \in S: x \prec_1 y \iff \map \pi x \prec_2 \map \pi y$

Then $\pi$ is an injection.

Proof
$\pi$ is not an injection.

Hence:
 * $\exists x, y \in S: \map \pi x = \map \pi y$

As $S$ is strictly totally ordered:
 * $x \prec_1 y$ or $y \prec_1 x$

, let $x \prec_1 y$.

Then we have:
 * $\map \pi x = \map \pi y$

But :
 * $\map \pi x \prec_2 \map \pi y$

Because $\prec_2$ is a strict ordering, it follows that:
 * $\map \pi x \ne \map \pi y$

It follows by Proof by Contradiction that it cannot be the case that $\pi$ is not an injection. Hence the result.