Characterization of Prime Element in Meet Semilattice

Theorem
Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice.

Let $p \in S$,

Then:
 * $p$ is prime element


 * for all non-empty finite subsets $A$ of $S$:
 * if $\inf A \preceq p$, then there exists element $x$ of $A$ such that $x \preceq p$.
 * if $\inf A \preceq p$, then there exists element $x$ of $A$ such that $x \preceq p$.