If we redirect permanently (using 301) from an old website to a new wesbite. In this case can we remove the old website? Will the redirected URLs from old to new website will still work or not? Or is it necessary to keep the old website in in place always?

Let me make it simple by the following example.

If I used this URL www.aabbcc.com for 4 years and all my incoming links are linked to it. Now I changed my URL to www.ccddeeff.com. I applied 301 redirect from www.aabbcc.com to www.ccddeeff.com. Currently, this scenario is working fine But what If I remove this page (or module) www.aabbcc.com permanently. Will the traffic still be redirected from www.aabbcc.com to www.ccddeeff.com? Or do i need to keep www.aabbcc.com for always?