Join Succeeds Operands

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Theorem

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

Let $a, b \in S$ admit a join $a \vee b \in S$.


Then:

$a \preceq a \vee b$
$b \preceq a \vee b$

That is, $a \vee b$ succeeds its operands $a$ and $b$.


Proof

By definition of join:

$a \vee b = \sup \set {a, b}$

where $\sup$ denotes supremum.


Since a supremum is a fortiori an upper bound:

$a \preceq \sup \set {a, b}$
$b \preceq \sup \set {a, b}$

as desired.

$\blacksquare$


Also see