Join in Inclusion Ordered Set

Theorem
Let $P = \left({X, \subseteq}\right)$ be an inclusion ordered set.

Let $A, B \in X$ such that
 * $A \cup B \in X$

Then $A \vee B = A \cup B$

Proof
By Set is Subset of Union:
 * $A \subseteq A \cup B$ and $B \subseteq A \cup B$

By definition:
 * $A \cup B$ is upper bound for $\left\{ {A, B}\right\}$

We will prove that
 * $\forall C \in X: C$ is upper bound for $\left\{ {A, B}\right\} \implies A \cup B \subseteq C$

Let $C \in X$ such that
 * $C$ is upper bound for $\left\{ {A, B}\right\}$.

By definition of upper bound:
 * $A \subseteq C$ and $B \subseteq C$

Thus by Union is Smallest Superset:
 * $A \cup B \subseteq C$

By definition of supremum:
 * $\sup \left\{ {A, B}\right\} = A \cup B$

Thus by definition of join:
 * $A \vee B = A \cup B$