Leibniz's Law for Sets

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Theorem

Let $S$ be an arbitrary set.


Then:

$x = y \dashv \vdash x \in S \iff y \in S$

for all $S$ in the universe of discourse.


This is therefore the justification behind the notion of the definition of set equality.


Proof

A direct application of Leibniz's law.


Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.


Sources