Working on set theory I stumbled across this exercise:
Let $\{A_\alpha: \alpha \in \mathcal{A}\}$ be a family of well-ordered sets. If for every pair of sets, one is an ideal of the other, prove that in $\bigcup_\alpha A_\alpha$ there is exactly one well-ordering that coincides with that on each $A_\alpha$.
I understand it intuitively but I am not quite sure if my attempt to prove it is correct.
Existence
First, it is known that the set of all ideals of a given set (in this case $\bigcup_\alpha A_\alpha $) is well-ordered by inclusion. And by assumption, given two ideals of the form $A_{\lambda}$, $A_{\mu}$ one is an ideal of the other. Thus we can form a particular (some may be missing but it is immaterial) chain of the given ideals, $$ A_{\alpha_1} \subset A_{\alpha_2} \subset A_{\alpha_3} \subset... \qquad. $$
Extending the ordering in a natural way, we obtain a new one which coincides with that on each $A_{\alpha_i}$.
Let $B\neq \emptyset$ be a subset of $\bigcup_\alpha A_\alpha $, we must show that that there is a first element of $B$. For, we choose $A_{k}$ to be the first ideal (in the sense of inclusion) in which any element of $B$ lies (this is feasible beacuse we already formed a chain -well ordered- of possible sets). Since $B$ is well ordered, the subset of elements in $B$ which are also in $A_k$, that is $A_k\bigcap B$ (nonepty), has a first element, say $b_0$. It is evident that $b_0\leq a$ for any other $a$ in the union, meaning we have found a least element and proved well-ordering.
Uniqueness
If there were two different orderings $\leq_1, \leq_2$, they should differ in at least a pair of elements, $a\leq_1b$ and $b\leq_2 a$ for $a\neq b$. Let $A_\beta$ be the first set which contains both $a$ and $b$. The order we constructed above was a well ordering that coincided with that of $A_\beta$ when working in such set. If we consider the set $\{a, b\}$, subset of $A_\beta$, which was well ordered, we see that one (and only one) of $a$, $b$ must be a first element. That means different orders cannot hold the requiered properties at the same time.
Thanks in advance.