The Fundamental Theorem of Hopf Modules: Every Hopf module over a Hopf algebra is trivial.

Several of the classical proofs of major results in Hopf algebras involve constructing/defining a Hopf module such that the theorem desired holds precisely when this Hopf module is trivial. So the non-existence of a non-trivial such object is quite important.

The concept of Hopf module has of course been generalized (in many different ways), though their triviality is no longer guaranteed in such broader contexts.