Is it unlawful to take an email list from a company that you work for and then use that email list at their new company.

It is not unalwful but unethical and embarrassing to the new company.
First of all, before terminating that employee, it should be clear to him what are do's and don't upon his exit. His username should be also blocked to prevent these things.

Yes it is unlawfui to take an email list from a former employer. That is theft. It would be the same thing if the employee took a mailing list of all the company's clients. Theft is theft and is punishable by law.

Is it unlawful to take an email list from a company that you work for and then use that email list at their new company.

Yes, it's theft. More and more people are being charged with internet stealing and this is one of the ways it's done.

Email list from a former employer. That is theft. It would be the same thing if the employee took a mailing list of all the company's clients.