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 elements^{1}, 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
Please note that repeating the same elements in a set determines the same set. ↩