NORAD, it seems, believe Father Christmas exists. So here's a proof of his existence, that I found in Logics, Categories, and Colimits for Artificial Intelligence: Exercise Sheet 9 by Till Mossakowski of the University of Freiburg.
To prove that Father Christmas exists, we'll assume the opposite and show that it leads to a contradiction. So assume that Father Christmas does not exist:
But then we can apply the rule of existential-quantifier introduction. This is the formal equivalent of inferring from John is fat that someone is fat, i.e. that there exists an x such that x is fat. So we infer from Father Christmas does not exist that there exists an x such that x does not exist:
But (2) is a contradiction.
Therefore, our assumption (1) must be wrong.
Therefore, Father Christmas exists. ∎
Of course, there could be an unnoticed bug in this proof technique. So before posting, I tested it on the existence of: (a) wasps; (b) winter; (c) Windows 7. Wasps are something I've never seen the point of, but they exist, and so do the other things. When tested with each, the proof technique ran to completion in my brain, and gave a result that's consistent with reality. Therefore, I claim it adequately tested. Any programmer will tell you that three test runs is enough.