What is a finite set? It is a set such has no proper subset has a bijection with the full set. Use that fact and prove formally the principle.
How many proofs are there for the pigeonhole principle? I know of one proof that just uses composition of functions to develop lemmas, and then ultimately come up with the Pigeonhole principle. The proof starts with the basic problem of 2 different ways of counting sets (i.e if and are bijections with the same codomain, then ). Namely, it says the following: Suppose that and are finite, non-empty sets such that . Then is not an injection.
There are several proofs as you observe. This of is found in many discrete mathematics texts.
Given that , recall that .
Suppose that .
Using that we have that is a contradiction.
Thus .
The value of this proof is that it also proves the generalized form. If we have m pigeons and n pigeonholes with the some hole contains at least pigeons (that is the ceiling function).
To see this proof use the assumptions above, using the supposition that .
This time that is a contradiction.
Thus .