This book I'm reading contains the following claim:
$(\Omega,A)$ is a sigma algebra iff it is a $\Pi$ system and a Dynkin System.
Proving the forward direction, its easy to show that if $(\Omega,A)$ is a sigma algebra then it is a Dynkin System.
However, its not true though that every sigma algebra is a $\Pi$ system. For instance if: $\Omega = \{1,2,3\}$ and $A = \{ \emptyset, \{1\},\{3\},\{2,3\},\{1,2\},\{1,2,3\}$}. Evidently $\{2,3\}\cap \{1,2\} \notin A$.
Is the forward direction true then?