Continuous iff For Every Element There Exists Ideal Element Precedes Supremum

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

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 $L: x \preceq \sup J \implies I \subseteq J$

Sufficient Condition
Let $L$ be continuous.

By Continuous iff Way Below Closure is Ideal and Element Precedes Supremum:
 * $\forall x \in S: x^\ll$ is an ideal in $L$ and $x \preceq \sup \left({x^\ll}\right)$ and
 * for every ideal $I$ in $L$: $x \preceq \sup I \implies x^\ll \subseteq I$

where $x^\ll$ denotes the way below closure of $x$.

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

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