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.
Also see
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.