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][1] 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.


  [1]: https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem