We define $ω$ to be the set of natural numbers, i.e. $ω=$ $\cap$ {$x$ | $0\in x \wedge \forall u\in x$ $ u+1\in x $}
Accordingly, I have managed to show that $ω \subset\text{Ord}$, where $\text{Ord}$ is the class of all ordinal numbers.
Since I was asked to prove that $\omega \in \text{Ord}$, it is only left to prove that $\omega$ itself is transitive (due to definition which states:
$x \in \text{Ord}$ iff x is transitive and every element of $x$ is transitive too.)
Does anyone have any idea of how to prove transitivity of $\omega$, where by transitivity I mean, if $x \in \omega$, then $x \subset \omega$...
Under the von Neumann construction of the natural numbers, which is needed for this to be true with $\omega$ as defined, you take $0 = \varnothing$ and $n+1 = n \cup \{ n \}$ for all $n < \omega$. You can prove by induction on $n$ that, for all $x$, you have $x \in n$ if and only if $x \in \omega$ and $x < n$; the result then follows fairly easily.