Meet Precedes Operands

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

Let $a, b \in S$ such that their meet $a \wedge b \in S$.

Then:


 * $a \wedge b \preceq a$
 * $a \wedge b \preceq b$

i.e., $a \wedge b$ precedes its operands $a$ and $b$.

Proof
By definition of join:


 * $a \wedge b = \inf \left\{{a, b}\right\}$

where $\inf$ denotes infimum.

Since an infimum is a fortiori a lower bound:


 * $\inf \left\{{a, b}\right\} \preceq a$
 * $\inf \left\{{a, b}\right\} \preceq b$

as desired.