Characterization of Prime Ideal by Finite Infima

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

Let $I$ be an ideal in $L$.

Then
 * $I$ is a prime ideal


 * for all non-empty finite subset $A$ of $S: \left({ \inf A \in I \implies \exists a \in A: a \in I}\right)$

where $X$ is subset of $S$.

Sufficient Condition
Let $I$ be a prime ideal.

Define $\mathcal P\left({X}\right) :\equiv X \ne \varnothing \land \inf X \in I \implies \exists x \in X: x \in I$

where $X$ is subset of $S$.

Let $A$ be a non-empty finite subset of $S$.

By definition of empty set:
 * $\mathcal P\left({\varnothing}\right)$

We will prove that
 * $\forall x \in A, B \subseteq A: \mathcal P\left({B}\right) \implies \mathcal P\left({B \cup