Ordered Subset of Ordered Set is Ordered Set

Theorem
Let $L = \struct {S, \preceq}$ be an ordered set.

Let $\struct {S', \preceq'}$ be an ordered subset of $L$.

Then $\struct {S', \preceq'}$ is an ordered set.

Proof
By definition of ordered subset:
 * $S' \subseteq S$

Reflexivity
Let $x \in S'$.

By definition of subset:
 * $x \in S$

By definition of reflexivity:
 * $x \preceq x$

Thus by definition of ordered subset:
 * $x \preceq' x$

Transitivity
Let $x, y, z \in S'$ such that:
 * $x \preceq' y$ and $y \preceq' z$

By definition of ordered subset:
 * $x \preceq y$ and $y \preceq z$

By definition of transitivity:
 * $x \preceq z$

Thus by definition of ordered subset:
 * $x \preceq' z$

Antisymmetry
Let $x, y \in S'$ such that:
 * $x \preceq' y$ and $y \preceq' x$

By definition of ordered subset:
 * $x \preceq y$ and $y \preceq x$

Thus by definition of antisymmetry:
 * $x = y$