Characterization of Pseudoprime Element by Finite Infima

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a continuous lattice.

Let $p \in S$ be a pseudoprime element.

Let $A$ be a non-empty finite subset of $S$ such that
 * $\inf A \ll p$

where $\ll$ denotes the way below relation.

Then $\exists a \in A: a \preceq p$

Proof
By definition of pseudoprime element:
 * there exists a prime ideal $P$ in $L$: $p = \sup P$

By definition of way below closure:
 * $\inf A \in p^\ll$

By definition of reflexivity:
 * $p \preceq \sup P$

By Continuous iff Way Below Closure is Ideal and Element Precedes Supremum:
 * $p^\ll \subseteq P$

By definition of subset:
 * $\inf A \in P$

Thus by Characterization of Prime Ideal by Finite Infima
 * $\exists a \in A: a \in P$

By definition of up-complete:
 * $P$ admits a supremum.

By definition of supremum:
 * $p$ is upper bound for $P$.

Thus by definition of upper bound:
 * $a \preceq p$