How about a proof that, given a set, there always exists a bigger set?
For every set A, there is a set that doesn't inject into A. Take the set B of ordinals that inject into A. B is an ordinal and is not in B.
I think one can also fit a proof of Sylvester's theorem in 137 characters.
Take n points not on a line. Let L be a line containing >1 points minimizing the distance to a point off L. It contains exactly 2 points.