Another way of proving that something is nonzero is to prove that it is odd. One good example of that idea is the proof of [Sperner's lemma][1]



More generally, one can prove that something is nonzero by proving that it is nonzero mod p. 
That is the idea used in [Chevalley-Warning theorem][2] (one proves that the number of solutions is 0 mod p, then proves there is a trivial solution), and in the proof of [Cauchy's theorem][3].


  [1]: http://en.wikipedia.org/wiki/Sperner%2527s_lemma
  [2]: http://en.wikipedia.org/wiki/Chevalley-Warning_theorem
  [3]: http://en.wikipedia.org/wiki/Cauchy%2527s_theorem_%2528group_theory%2529