Successor is Supremum

Theorem

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

Let $x, y \in S$.

Then:

$x \preceq y$ if and only if $y = \map \sup {x, y}$

where $\map \sup {x, y}$ is the supremum of $x$ and $y$

Proof

Necessary Condition

Let $x \preceq y$.

By Ordering Axiom $(1)$: Reflexivity:

$y \preceq y$

Hence $y$ is an upper bound of $\set {x, y}$ by definition.

Let $z$ be an upper bound of $\set {x, y}$.

By definition of upper bound:

$y \preceq z$

Hence $y$ is the supremum of $x$ and $y$ by definition.

$\Box$

Sufficient Condition

Let $y$ be the supremum of $x$ and $y$.

Then $x \preceq y$ by definition of a supremum.

$\blacksquare$