This is Exercise III.2.20 of Bourbaki's Set Theory.
(Von Neumann ordinals are actually called "pseudo-ordinals" by Bourbaki, but I simply call them ordinals here)
Let $X$ be a transitive set, and suppose that each $x\in X$ is an ordinal. Then $X$ is an ordinal (Hint: for each $x\in X$, $x\cup\{x\}$ is an ordinal contained in $X$).
This statement is proven in many textbooks. The problem is that none of them uses Bourbaki's definition of ordinal:
For Bourbaki, a set $X$ is an ordinal if every proper transitive subset of $X$ is an element of $X$. [Edit: I'm not sure if this implies e.g. that $X$ is well-ordered by $\in$, since Bourbaki has no Axiom of Foundation.]
A proof that this implies one of the usual definitions (e.g. $X$ is a transitive set whose members are transitive) would be enough, too. [Sorry!]
This should be easy, but I don't know where to start. I'm glad for any help.
Second Edit: The statement I want to show in a hopefully clearer form:
Let $X$ be a transitive set such that any $x\in X$ has the property that any proper transitive subset of $x$ is an element of $x$. Then $X$ has the same property.