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Proof: (related to "Uniqueness of the Empty Set")

  • For any two given non-empty sets $X,Y$ assume, their elements $x\in X,y\in Y$ are sets themselves.
  • By the extensionality principle they are equal $x=y$ if they have the same elements.
  • This argument can be repeated for the elements of these elements (if any). We can repeat this argument for all elements of elements of elements, etc.
  • At some point, we might arrive at elements $x\in X$ and $y\in Y$ which do not contain any elements. Note that the existence of such empty elements $x=\emptyset$ and $y=\emptyset$ is ensured by the axiom of empty set.
  • Now, the extensionality principle says again, that $x=y$ are equal, even though $x=\emptyset$ and $y=\emptyset$.
  • In other words, all empty sets are not distinguishable. Thus, $\emptyset$ is unique.

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