Transitivity of Element of

A ε B means the set A is a element of the set B, right? That's not the same thing as A being a subset of B. You have sets that are elements of other sets. Try to construct a counterexample.

 

Pick A={a}. Pick B={A}={{a}}. So A ε B. Correct so far? Pick C={B}={{{a}}}. So B ε C. Is A ε C?? This is a little tricky. a is not equal to the set consisting of a.

 

You seem to have deleted the question. I thought I suggested you try to prove that A ε B and B ε C doesn't necessarily imply A ε C? Since inclusion is not transitive?

 

I think you are reaching way too far for a contradiction. C ε C puts you squarely in Bertrand-Russell paradox territory. A simple example of A ε B and B ε C with simple sets and A not an element of C will serve nicely. I basically gave you one. Follow it up. It shows inclusion isn't transitive. That's all you need.

 

I don't think so. Inclusion ISN'T transitive. You can't say A ε B and B ε C implies A ε C. At all.

 

C ε C violates regularity. But the statement C ε C doesn't follow from anything you've said before because inclusion isn't transitive.

 

Mmm. I'm not all that hot with set axiomatics. But you can simplify that. Suppose A ε B and B ε A, can you show that violates regularity?? Like I say, I don't have ZFC axioms at my fingertips.