Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication/Proof 2

Theorem
Let $\struct {S, \preceq_1}$ be a totally ordered set and let $\struct {T, \preceq_2}$ be an ordered set.

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

Then $\phi$ is an order embedding.

Proof
Let $\phi$ be strictly increasing.

Let $\map \phi x \preceq_2 \map \phi y$.

As $\struct {S, \prec_1}$ is a strictly totally ordered set:
 * Either $y \prec_1 x$, $y = x$, or $x \prec_1 y$.

that $y \prec_1 x$.

By the definition of a strictly increasing mapping:


 * $\map \phi y \prec_2 \map \phi x$

which contradicts the fact that $\map \phi x \preceq_2 \map \phi y$.

Therefore $y \nprec_1 x$.

Thus $y = x$, or $x \prec_1 y$, so $x \preceq_1 y$.

Hence:


 * $\map \phi x \preceq_2 \map \phi y \iff x \preceq_1 y$

and $\phi$ has been proved to be an order embedding.