Order Embedding is Injection

Theorem
Let $$\left({S; \preceq_1}\right)$$ and $$\left({T; \preceq_2}\right)$$ be posets.

Let $$\phi: \left({S; \preceq_1}\right) \to \left({T; \preceq_2}\right)$$ be an order monomorphism, i.e.:


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

Then $$\phi$$ is an injection.

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

We have:

$$ $$ $$ $$

So $$\phi$$ is an injection.