Today while day dreaming I discovered something interesting. I can prove a=a. Here's how: You can prove P=>P using natural deduction rules.(=> Intro) So you can prove that x is an element of a => x is an element of a Hence a is subset of a, and vice versa. By ZFC axiom of extension, a=a So a=a need not be an axiom, because it can be proven. In this sense, equality is not the fundamental concept. Set membership is.