Join Succeeds Operands

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.

$\blacksquare$