Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing/Reverse Implication/Proof 1

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 $x \preceq_1 y$.

Then $x = y$ or $x \prec_1 y$.

Let $x = y$.

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

so:
 * $\phi \left({x}\right) \preceq_2 \phi \left({y}\right)$

Let $x \prec_1 y$.

Then by the definition of strictly increasing mapping:
 * $\phi \left({x}\right) \prec_2 \phi \left({y}\right)$

so by the definition of $\prec_2$:
 * $\phi \left({x}\right) \preceq_2 \phi \left({y}\right)$

Thus:
 * $x \preceq_1 y \implies \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$

It remains to be shown that:
 * $\phi \left({x}\right) \preceq_2 \phi \left({y}\right) \implies x \preceq_1 y$

Suppose that $x \npreceq_1 y$.

Since $\preceq_1$ is a total ordering:
 * $y \prec_1 x$

Thus since $\phi$ is strictly increasing:
 * $\phi \left({y}\right) \prec_1 \phi \left({x}\right)$

Thus:
 * $\phi \left({x}\right) \not\preceq_1 \phi \left({y}\right)$

Therefore:
 * $x \npreceq_1 y \implies \phi \left({x}\right) \npreceq_2 \phi \left({y}\right)$

By the Rule of Transposition:
 * $\phi \left({x}\right) \preceq_2 \phi \left({y}\right) \implies x \preceq y$