Successor is Supremum
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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.
$\blacksquare$