I am studying a course on ZF Set Theory and have recently been considering whether or not $X$ being well orderable implies that $X \times X$ is well orderable.
More formally my question is the following:
Prove: “$X$ being well orderable implies that $X \times X$ is also well orderable”
It seems obvious to conclude that the answer here is the affirmative and intuitively it is easy to see why this is the case (where the minimal element of $X \times X$ will be the element $(a,a)$ where $a$ is the minimal element of $X$ and each subset of $X \times X$ will have a minimal element which will be the ordered pair $(c,d)$ where $c$ is the minimal element of the first subset of $X$ and $d$ is the minimal element of the second subset of $X).$
However, constructing a proof of the claim is where I am having difficulties. Is there a suitable reference or proof available to make my intuition more rigorous?
As suggested in the comments, this can be achieved using the lexicographic ordering, however, I would still be interested in seeming a formalisation of this.
Thank you in advance.