transitive property within the sufficient and necessary lesson

This is regarding the A---B---C---D---E chain. If you say that C exists, we can safely conclude that D and E exist; however, we can't conclude for sure that A and B do. But, I wonder, if all Bs are Cs, wouldn't that mean that some Bs are Cs and therefore some Cs are Bs? If so, could we conclude that some Bs and some As do in fact exist?