Use
$db->user['result'][0]
By the way, array keys that aren't quoted strings are generally bad practice (and throw a warning or notice, I can't remember which). So, when you access the first name, use this instead:
$db->user['first_name']
Why? Off to the PHP documentation:
This [using array keys unquoted] is wrong, but it works. The reason is that this code has an undefined constant (bar) rather than a string ('bar' - notice the quotes). PHP may in future define constants which, unfortunately for such code, have the same name. It works because PHP automatically converts a bare string (an unquoted string which does not correspond to any known symbol) into a string which contains the bare string. For instance, if there is no defined constant named bar, then PHP will substitute in the string 'bar' and use that.
In addition, there is a performance degradation if you don't quote your array keys; PHP must lookup the token in the constants table, then, realising it isn't defined, replace it with its equivalent string.