Although you may no longer have access to mail sent to the old email address, that address and its associated password still function as a valid Apple ID.
See also FAQ at http://support.apple.com/kb/HE37
Q: I no longer have an email address that was also my Apple ID. Can I still use the email address as my Apple ID?
A: Apple recommends you change your Apple ID to your current, working email address. This will not create another Apple ID, it will only change it to your working email address. See How to change the name you use for your Apple ID if you'd like more information.