Characterization of Prime Element in Inclusion Ordered Set of Topology

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $L = \left({\tau, \preceq}\right)$ be an inclusion ordered set of $\tau$.

Let $Z \in \tau$.

Then $Z$ is prime element in $L$
 * $\forall X, Y \in \tau: X \cap Y \subseteq Z \implies X \subseteq Z \lor Y \subseteq Z$

Sufficient Condition
Assume that
 * $Z$ is prime element in $L$.

Let $X, T \in \tau$ such that
 * $X \cap Y \subseteq Z$

By Join and Meet in Inclusion Ordered Set of Topology and definition of inclusion ordered set:
 * $X \wedge Y \preceq Z$

By definition of prime element:
 * $X \preceq Z$ or $Y \preceq Z$

Thus by definition of inclusion ordered set:
 * $X \subseteq Z$ or $Y \subseteq Z$

Necessary Condition
Assume that L$\forall X, Y \in \tau: X \cap Y \subseteq Z \implies X \subseteq Z \lor Y \subseteq Z$

Let $X, Y \in \tau$ suvh that
 * $X \wedge Y \preceq Z$

By Join and Meet in Inclusion Ordered Set of Topology and definition of inclusion ordered set:
 * $X \cap Y \subseteq Z$

By assumption:
 * $X \subseteq Z$ or $Y \subseteq Z$

Thus by definition of inclusion ordered set:
 * $X \preceq Z$ or $Y \preceq Z$