Preceding iff Meet equals Less Operand

Theorem
Let $\left({S, \preceq}\right)$ 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:
 * $x \wedge y = \inf \left\{ {x, y}\right\}$

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

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

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

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

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