Characterization of Prime Element in Inclusion Ordered Set of Topology

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $L = \struct {\tau, \preceq}$ 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 $X, Y \in \tau$ such 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 $Y \preceq Z$