Prove that their is one empty set? (I am aware this question has been asked before but mine is little different). Applying the axiom of extensibility , I am able to prove this, but still I am not able to understand the significance of this result, it appears to me that this is true for every set. Say $A=(a_1,a_2,a_3)$ then $A=B$ iff $B=(a_1,a_2,a_3)$ hence their is unique set $A$ (that is $A$ itself).
 A: The axiom of extensionality states that two sets are equal if and only if they have the same elements. In particular, if we have two sets $x,y$ without any elements, then $x = y$ and thus, there is a unique empty set.
If we drop the axiom of extensionality, it is possible to have sets $U,V$ such that $U \neq V$ and neither $U$ nor $V$ contain any element. These sets can be regarded as "urelements" (that are also called "atoms") - a concept that has been of significant interest in the earlier days of set theory. In fact, Fraenkel and Mostowski used urelements in independence arguments long before Cohen invented forcing. One important result is that ZF + "urelements" doesn't imply the axiom of choice and so ZF - extensionality doesn't imply the axiom of choice. The fact that extensionality has to fail in order to apply the methods of Fraenkel and Mostowski is a considerable weakness of their theory and for several decades, no one saw how to adapt their methods in order to construct new models of ZF. In fact, no one was able to prove the consistency of $\neg V = L$, relative to ZF.
In 1963, with Cohen's introduction of forcing, all of this changed and soon he realized that his theory could be used to imitate Fraenkel's and Mostowski's methods without violating extensionality. A development that lead to the theory of "permutation models".
TL;DR Having a unique empty set is - in a sense - the requirement that our models don't have urelements and this complicates the early independence results of Fraenkel and Mostowksi. 
