Why can you still use a disposed object?

I'm battling a bit to understand disposing of objects and garbage collection. In particular, I don't understand why I can still use a disposed object. I'm not trying to do anything practically, I'm just playing around with theory at the moment and in my understanding, I thought that I wouldn't be able to do the following:

class Program
{
    static void Main(string[] args)
    {
        Person p = new Person();
        using (p)
        {
            p.Name = "I am Name";
        }
        Console.WriteLine(p.Name); // I thought this would break because I've already disposed of p
        Console.ReadLine();
    }
}
public class Person : IDisposable
{
    public string Name;
    public void Dispose()
    {
        Console.WriteLine("I got killed...");
    }
}

I'm hoping that someone could perhaps give me some direction or guidance here so as to clear up my misunderstanding of this concept?