But that is not what i meant, axioms can rely on other axioms without it being possible to prove the newly defined axiom from the older one. For instance, uniformity of nature first assumes nature exists. (if you want to call those postulates, very well, we'll talk postulates then.)
It may be possible to turn sets of axioms depending(not provable from) on eachother into sets that do not. But it may be difficult to do so, and hinder discussion. I do not see axioms depending on(/but not provable from) eachother as a problem.
About "A=> not A is a proof that A is false", i have not yet seen that problem here yet. What i meant by saying that, is that there was unnessesary redundancy, although now i see the definition was only the above six lines of quoted link. I think that we should use wikipedias axiom instead.

hope we can back to axioms themselves