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](https://ncatlab.org/nlab/show/Hopf+module) (in many different ways), though their triviality is no longer guaranteed in such broader contexts.