Let $\alpha$ and $\beta$ be ordinals. Show that $Z:=(\alpha \times \{0\}) \cup (\beta \times \{1\})$ is well-ordered, for the order
$$(\lambda, i) \leq (\mu,j) \iff (i< j) \lor((i=j) \land(\lambda \leq \mu))$$
Well, I guess I will have to show that if $\emptyset \neq Y \subseteq Z$, then $Z$ has a minimum.
How would I construct such a minimum?
I thought maybe to look at the sets
$$Z_\alpha:=\{\gamma \in \alpha: (\gamma,0) \in Y \mathrm{\ or\ } (\gamma,1) \in Y\}$$ $$Z_\beta:=\{\gamma \in \beta: (\gamma,0) \in Y \mathrm{\ or\ } (\gamma,1) \in Y\}$$
At least one of these is non-empty. But then I have to split up multiple cases and this does not seem elegant, so I was hoping there is a simple way to see this is well-ordered.
It is easy to see that $Z$ is total, so we can also show that $Z$ has no strictly decreasing sequence.
Thanks in advance.
Hint: the order is just a copy $A = \alpha \times \{0\}$ of $\alpha$ followed by a copy $B = \beta \times \{1\}$ of $\beta$. Given $\emptyset \neq Y \subseteq Z = A \cup B$, there are two cases to consider: (1) $Y \cap A = \emptyset$, and (2) $Y \cap A \neq \emptyset$. In case (2), use the fact that $A$ is well-ordered. In case (1), what can you say about $ Y \cap B$?