Correct. The Berne Convention did away with 'formalities' like having to register your copyrights in order to hold them. You still need to have a registered copyright to sue for statutory damages in the USA, though.
IANAL but I believe you are correct although you get some extra legal protections by including the notice (which only need include the word or symbol and the name and year). Namely, an infringer cannot claim they did so accidentally... or something like that. FWIW.
