I just read a proof and, after struggling some time with a mental leap, I think that it uses tacitly the following:


Let $\kappa$ be a regular cardinal, $\theta > \kappa$ a regular cardinal too then:
$ S \subset \kappa$ is stationary if and only if 
$\forall \mathcal{A} = (H(\theta), \in, <,..) \exists M \prec \mathcal{A}, |M| < \kappa,$ such that $sup(M \cap \kappa) \in S$.


Now my questions are:


1. Is this statement above even true? (I think so as I have a proof, but this doesn't have to mean anything)
2. It appears to me that the latter part of this characterization is a quite strong assumption as $\mathcal{A}$ might contain a lot of additional information, so is there a possibility to weaken it? Or could you mention any similar statements to the one above?


Thank you


EDIT: I accepted the answer of Philip, simply because he has lower points. Francois answer would have deserved it too.



A characterization of stationarity?
(I first wanted to give an answer, but I was not quick enough. I then wanted to add a small comment and found out after 20 minutes that I had insufficient reputation.)


The comment was regarding 2) of oktan's original query: having $H(\theta)$
in the structure is overkill: it suffices to have $( \kappa, <, \in, C)$. (One does not need the structure to be able to express $C$ is closed.'')


