Why are all von Neumann ordinals contained in each other?

Observe that for given ordinals $A$ and $B$ the following claims hold true:

• If $A$ is a proper subset of $B$, then $A \in B$.
• $A \cap B$ is an ordinal.

Now, given ordinals $A \neq B$ consider $A \cap B \subseteq A,B$. If $A \cap B = A$, then $A \subseteq B$, so that either $A = B$ or $A \in B$. Analog $A \cap B = B$ implies either $B = A$ or $B \in A$. If $A \cap B$ is a proper subset of both $A$ and $B$, then $A \cap B \in A \cap B$, which contradicts the axiom of regularity.