Order Embedding is Injection

Theorem
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

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

That is:


 * $\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

Then $\phi$ is an injection.

Proof
Suppose $\phi: S \to T$ is a mapping such that:
 * $\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

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

So $\phi$ is an injection.