Meet Irreducible iff Finite Infimum equals Element

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

Let $x \in S$.

Then
 * $x$ is meet irreducible


 * for every non-empty finite subset $A$ of $S$: $x = \inf A \implies x \in A$

Necessary Condition
Assume that
 * for every non-empty finite subset $A$ of $S$: $x = \inf A \implies x \in A$

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

By definition of meet:
 * $x = \inf \left\{ {y, z}\right\}$

By assumption:
 * $x \in \left\{ {y, z}\right\}$

Thus by definition of unordered tuple:
 * $x = y$ or $x = z$