Finite set
Uniqueness of cardinality

Proof: If X {\displaystyle X} is empty (has cardinality 0), then there cannot exist a bijection from X {\displaystyle X} to any nonempty set Y, {\displaystyle Y,} since nothing mapped to y 0 ∈ Y. {\displaystyle y_{0}\in Y.} Assume, by induction that the result has been proven up to some cardinality n. {\displaystyle n.} If X, {\displaystyle X,} has cardinality n + 1, {\displaystyle n+1,} assume it also has cardinality m. {\displaystyle m.} We want to show that m = n + 1. {\displaystyle m=n+1.} By the lemma above, X − { x 0 } {\displaystyle X-\{x_{0}\}} must have cardinality n {\displaystyle n} and m − 1. {\displaystyle m-1.} Since, by induction, cardinality is unique for sets with cardinality n, {\displaystyle n,} it must be that m − 1 = n, {\displaystyle m-1=n,} and thus m = n + 1. {\displaystyle m=n+1.}
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