Can a PhD supervisor take unpublished ideas from a student's dissertation? Rephrased question: Can a PhD supervisor take unpublished ideas from a student's dissertation and work on them hence denying the student the opportunity to work on her ideas herself?

I completed a PhD and took on a short postdoc immediately afterwards with the same supervisor. The idea was to be doing something while I apply for bursaries or funds to continue research on the ideas I proposed in my thesis. Throughout my PhD I had minimal supervision and minimal input of any sort from him. In fact, in 5 years, he never proposed or even simply put me on track to an idea or a solution. I know for a fact that my supervisor barely read a fourth of my dissertation; however, I had spoken to him about my ideas several times and they are mentioned in my dissertation as my own. I know he is aware of that because I sent him the acknowledgement part by email before putting it in my thesis. 

Now he decided to take my ideas for himself and got a new student to work on them with the contacts I developed myself because he was incapable of doing that himself. I supervised the new student who is working on my ideas for a particular application and got him going etc and I will be on the eventual paper. However, my supervisor has now decided to apply my idea to another application which I proposed myself in my thesis as well and given that I am leaving soon, I’ve been told that I will not be on that paper because I haven’t contributed directly. Beside the fact that this is still an application of my idea and one that I suggested myself in my thesis, the supervision I’ve given to the student is perfectly applicable to this idea as well because they are related. 

Can my supervisor do this even though he knows I want to work on them myself? Can I stop him from doing this? Frankly, I’m gutted and totally demoralised. There is no way for me to compete with him and get a paper out before him.
A PhD dissertation is published work, not a private document. It is publicly available to anyone with a library access (via interlibrary loan or via an appropriate dissertation vendor).

Of course, if anyone (including your former PhD supervisor) uses results from the dissertation, they must include an appropriate citation. That way you get a credit for your contribution. 

If your contribution is considered valuable by the standards of your field, you can write it up and publish it in a refereed journal for better visibility (compared to the dissertation). You can also upload your thesis on an institutional repository or arXiv.

My advice: do not waste time arguing about ownership of ideas. It will not help your career. You had one idea that was good enough to interest people; make more!