Preceding iff Meet equals Less Operand

Theorem
Let $\struct {S, \preceq}$ be a meet semilattice.

Let $x, y \in S$.

Then
 * $x \preceq y$ $x \wedge y = x$

Sufficient Condition
Let
 * $x \preceq y$

By definition of meet:
 * $x \wedge y = \inf \set {x, y}$

By definitions of lower bound and reflexivity:
 * $x$ is lower bound for $\set {x, y}$

and
 * $\forall z \in S: z$ is lower bound for $\set {x, y} \implies z \preceq x$

Thus by definition of infimum:
 * $x = \inf \set {x, y} = x \wedge y$

Necessary Condition
Let
 * $x \wedge y = x$

Thus by Meet Precedes Operands:
 * $x \preceq y$