In a MathOverflow thread on "nuking mosquitos", Andrej Bauer offered the following proof:
If two elements in a poset have the same lower bounds then they are equal by Yoneda lemma.
I understand that a poset can be considered to be a category with at most one arrow between any two objects, and I understand the statement of the Yoneda lemma, although I have little experience in using it. But I do not understand this proof. How does the Yoneda Lemma help?
This is explained in my blog post on the Yoneda lemma.
By the way, I do not consider this argument "nuking mosquitos." The Yoneda lemma is hardly a nuke; I would reserve that term for a highly technical result which requires a long proof. The proof of the Yoneda lemma is extraordinarily short and elegant. Besides, even this seemingly trivial special case can be surprisingly useful.