Join Succeeds Operands

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Theorem

Let $\left({S, \preceq}\right)$ be an ordered set.

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


Then:

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

i.e., $a \vee b$ succeeds its operands $a$ and $b$.


Proof

By definition of join:

$a \vee b = \sup \left\{{a, b}\right\}$

where $\sup$ denotes supremum.


Since a supremum is a fortiori an upper bound:

$a \preceq \sup \left\{{a, b}\right\}$
$b \preceq \sup \left\{{a, b}\right\}$

as desired.

$\blacksquare$


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