# Leibniz's Law for Sets

## 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.