Events One of Which equals Union

Theorem
Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.

Let $A, B \in \Sigma$ be events of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$.

Let $A$ and $B$ be such that:
 * $A \cup B = A$

Then whenever $B$ occurs, it is always the case that $A$ occurs as well.

Proof
From Union with Superset is Superset:


 * $A \cup B = A \iff B \subseteq A$

Let $B$ occur.

Let $\omega$ be the outcome of $\EE$.

Let $\omega \in B$.

That is, by definition of occurrence of event, $B$ occurs.

Then by definition of subset:
 * $\omega \in A$

Thus by definition of occurrence of event, $A$ occurs.

Hence the result.