I've been trying to understand $\omega_1$ (the first uncountable ordinal) and I've seen definitions but never a proof that it exists so I tried to come up with my own. (Apologies if this is basic, but I couldn't find a proof online).
Background (since people use different notations and definitions):
Transitive Set: a set $X$ is transitive if every element of $X$ is a subset of $X$.
Ordinal: an ordinal is a set that is: (1) totally ordered by $\in$; and (2) transitive.
Claim: $\omega_1$, defined as the set of all countable ordinals, is an uncountable ordinal.
Proof (I hope): need to show that $\omega_1$ is an ordinal and uncountable.
Ordinal: (1) $\omega_1$ is totally ordered by $\in$ since this is true of any set of ordinals.
(2) let $\alpha\in\omega_1$. $\alpha$ is a countable ordinal (by definition) and for all $\beta\in\alpha$, $\beta$ is also a countable ordinal (since $\beta\in\alpha\Rightarrow\beta\subset\alpha$). So $\beta\in\omega_1$.
Uncountable: now that we know $\omega_1$ is an ordinal, we know its uncountable since $\omega_1$ is defined as all the countable ordinals and if $\omega_1$ were countable, we would have $\omega_1\in\omega_1$ which we cannot have in ZFC. $\square$
Thanks in advance for any help.