Order Embedding is Injection

Theorem
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

Let $\phi: S \to T$ be an order embedding.

That is:


 * $\forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$

Then $\phi$ is an injection.

Proof
Suppose $\phi: S \to T$ is a mapping such that:
 * $\forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$

Then, for all $x, y \in S$:

So $\phi$ is an injection.