I'm having some trouble proving the following statement:
Any set $S$ of ordinal numbers is well ordered" which is equivalent (i think) to the statement "The class $On$ of all ordinals is well ordered".
The ordering i'm talking about is obviously $\in$. Here's where I'm stuck:
I've already proven that given $\alpha, \beta$ ordinals it is either $\alpha\in\beta$ or $\alpha=\beta$ or $\beta\in\alpha$ therefore $On$ is totally ordered. Now let $S$ be a nonempty subset of $On$; we want to prove that $S$ has a minimum element; Let $\beta=\bigcap S$. It is obvious that if $\gamma\in S$, then $\beta\subset\gamma$ and since $\beta$ is a transitive set (intersection of transitive sets) we have that $\beta$ is an ordinal. What i can't prove is that $\beta \in S$. What am I not seeing here?