Axiom: Axiom of Extensionality

If each element of the set \(X\) is also an element of the set \(Y\) and vice versa, then both are the same. In other words, a set is determined by its elements1, which is known as the extensionality principle.

\[\forall X~\forall Y (\forall z~(z\in X \Leftrightarrow z\in Y)\Rightarrow X=Y)\]


Explanations: 1

Axioms: 1 2 3 4
Corollaries: 5
Definitions: 6
Explanations: 7
Parts: 8
Proofs: 9 10 11 12 13 14 15 16 17 18 19

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  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
  2. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994


  1. Please note that repeating the same elements in a set determines the same set.