Given the definition of subsets and equality of sets:

• A is a subset of B if x is an element of A implies that x is an element of B for every set x.
• A = B if A is a subset of B and B is a subset of A

Why is it impossible to decide whether two circular sets I = {I} and J = {J} are equal.

I mean, the way is see it is that I is not an element of J, since only J is an element of J, so the two circular sets are not equal.

What's wrong in my reasoning?

Given the definition of subsets and equality of sets:

• A $\subset$ B, if x $\epsilon$ A $\rightarrow$ x $\epsilon$ B for every set x.
• A = B, if A $\subset$ B and B $\subset$ A

Why is it impossible to decide whether two circular sets I = {I} and J = {J} are equal.

I mean, the way is see it is that I is not an element of J, since only J is an element of J, so the two circular sets are not equal.

What's wrong in my reasoning?