Join Succeeds Operands

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 an upper bound:


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

as desired.

Also see

 * Meet Precedes Operands