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

Theorem
Let $\left({S, \preceq_1}\right)$ be a totally ordered set and let $\left({T, \preceq_2}\right)$ 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.

Suppose that $\phi \left({x}\right) \preceq_2 \phi \left({y}\right)$.

As $\left({S, \prec_1}\right)$ is a linearly ordered set:
 * Either $y \prec_1 x$, $y = x$, or $x \prec_1 y$.

Suppose for the sake of contradiction that $y \prec_1 x$.

By the definition of a strictly increasing mapping:


 * $\phi \left({y}\right) \prec_2 \phi \left({x}\right)$

which contradicts the fact that $\phi \left({x}\right) \preceq_2 \phi \left({y}\right)$.

Therefore $y \not \prec_1 x$.

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

Hence:


 * $\phi \left({x}\right) \preceq_2 \phi \left({y}\right) \iff x \preceq_1 y$

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