Events One of Which equals Intersection

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:
 * $A \cap B = A$

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

Proof
From Intersection with Subset is Subset:


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

Let $A$ occur.

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

Let $\omega \in A$.

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

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

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

Hence the result.