I am concerned with showing the least element existence in every subset of $\alpha\times\beta$. Below is my attempt. My teacher has used similar argument to show $\mathbb{N}\times\mathbb{N}$ is a well-ordered by some 'given' ordering.
QUESTION:
Let $\alpha,\beta$ be ordinals. Show that $\alpha\times\beta$ with the ordering
$(\gamma,\delta)\triangleleft(\lambda,\kappa)\iff[\gamma\cup\delta<\lambda\cup\kappa\vee(\gamma\cup\delta=\lambda\cup\kappa\space\land\space(\gamma<\lambda\vee(\gamma=\lambda\space\&\space\delta<\kappa)))]$ is well-ordered.
SOLUTION:
Now I show that for every non-empty subset $b$ of $\alpha\times\beta$ we have a least element. We proceed as follows:-
If $\emptyset\neq b\subseteq\alpha\times\beta$ , define $k$ to be the least ordinal in $\{x\cup y\mid (x,y)\in b\}$ with respect to ordering $\in_{Ord}$. The set $\{z\in {Ord}\mid (z,k\backslash z)\in b\}$ is non-empty and if $z_0$ is its least element (again w.r.t $\in_{Ord}$ then $(z_0,k\backslash z_0)$ is the $\triangleleft$ -least element of $b$.
Am I right?