# Ordered Subset of Ordered Set is Ordered Set

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## 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:

$x \preceq' x$

$\Box$

### 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:

$x \preceq' z$

$\Box$

### 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:

$x = y$

$\blacksquare$