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.

Also see

• Axiom:Axiom of Extensionality

Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.

Sources

• 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.): $\S 4.21$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Leibniz's law: 2.