Prime Element is Meet Irreducible

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

Let $p \in S$.

Let $p$ be a prime element of $L$.

Then $p$ is meet irreducible in $L$.

Proof
Let $p$ be a prime element.

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

By definition of reflexivity:
 * $x \wedge y \preceq p$

By definition of prime element:
 * $x \preceq p$ or $y \preceq p$

By Meet Precedes Operands:
 * $p \preceq x$ and $p \preceq y$

Thus by definition of antisymmetry:
 * $p = x$ or $p = y$