Prime is Pseudoprime (Order Theory)

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice.

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

Then $p$ is pseudoprime.

Proof
By Lower Closure is Prime Ideal for Prime Element:
 * $p^\preceq$ is prime ideal.

By Supremum of Lower Closure of Element:
 * $ \sup \left({ p^\preceq }\right) = p$

Hence $p$ is pseudoprime.