Facts or Theorems Similar to Pigeonhole Principle

Question

Are there any facts or theorems in mathematics having similar energy with pigeonhole principle? What I mean by similar energy is the statement is simple, "trivial" as you don't need to prove it to understand why it's true, but has many useful and powerful applications to prove many results. Thank you.

Answer 1

There is no integer (strictly) between zero and one. Looks pretty stupid, but many irrationality and transcendence proofs rely on it. [and in some sense it is equivalent to the pigeonhole principle]

Example: the proof of the transcendence of $e$ given in http://www.math.leidenuniv.nl/~evertse/dio15-4.pdf – note particularly the statement after Corollary 4.3, "Our aim is to show that for a suitable choice of $f$, the quantity $M := q_0F(0) +\cdots+ q_nF(n)$ is a non-zero integer with $|M| < 1$."

The proof of Lindemann-Weierstrass later in that same document is much more elaborate but also comes down to the non-existence of an integer $M$ with $0<M<1$.

This non-existence crops up again in the middle of the proof of the Gel'fond-Schneider Theorem, in the middle of page 64 of the same document.