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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?

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    The [silly] tag is... silly.2012-05-24
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    For some related examples in posts here, see the [floor function](http://math.stackexchange.com/a/147832/242) and [gcds and lcms.](http://math.stackexchange.com/q/147900/242)2012-05-24

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