You are not proving negatives. You do not prove that you don't have exact change for something, you prove what coins you have, and in doing so prove something contradictory to you have exact change.
You do not prove you do not have any more clean razors. You prove that all of your existing razors are dirty, and thus contradict that you have clean razors. Even then, all you are really proving is you haven't FOUND any clean razors that you have (you could have a box somewhere you forgot about)
You prove what is on your head, and since it is not your hat, you contradict that your hat is on your head.
The only way you can disprove something is to prove something else entirely that makes a claim impossible.
ie, you proved what coins you had, and that made it impossible that you had exact change.
There is no proof of positives and negatives, only proof and things contradicted by proof.
The Burden of Proof is also a lazy illogical principle. It allows one person to assume an inherint truth and force others to provide contradictory proof, all the while never haven proven the statement in the first place.
The illogic comes with the ease of turning the tables "Where is the burden of proof to show me god does exist" vs "Where is the burden of proof to show me god does not exist"
The correct answer is "There is no proof if god exists or not" until someone shows a proof that contradicts one of those statements by supporting the other.