If X is a quasi-projective variety over a field k and f is a birational endomorphism of X that is injective outside a closed subset of codimension at least 2, we prove that f is an automorphism. This generalizes an old theorem of Ax and proves a conjecture of Miyanishi from 2005. A key step in our proof is a finiteness result on class groups, which is of interest in its own right.