Continuous iff For Every Element There Exists Ideal Element Precedes Supremum

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

Then
 * $L$ is continuous


 * for every element $x$ of $S$ there exists ideal $I$ in $L$:
 * $x \preceq \sup I$ and for every ideal $J$ in $L: x \preceq \sup J \implies I \subseteq J$