Prime Element iff Meet Irreducible in Distributive Lattice

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice.

Let $p \in S$.

Then $p$ is a prime element $p$ is meet irreducible.

Proof
By Prime Element is Meet Irreducible:
 * if $p$ is a prime element, then $p$ is meet irreducible.

Assume that:
 * $p$ is meet irreducible.

Let $x, y \in S$ such that:
 * $x \wedge y \preceq p$

By definition of meet irreducible:
 * $p = p \vee x$ or $p = p \vee y$

Thus by Preceding iff Join equals Larger Operand:
 * $x \preceq p$ or $y \preceq p$