Every subgroup is subnormal implies normalizer condition

Verbal statement
If every subgroup of a group is subnormal, then the group satisfies the normalizer condition: it has no proper self-normalizing subgroup.

Proof
The proof follows from the observation that a proper subnormal subgroup is normal in a strictly bigger subgroup (namely, the one adjacent to it in its subnormal series), and hence, since every proper subgroup is subnormal, there cannot be a proper self-normalizing subgroup.