Ordering of Cardinality of Sets is Well-Defined

Theorem

The relation $\le$ in the context of cardinalities of sets is well-defined in the sense that:

if $\card A = \card {A'}$ and $\card B = \card {B'}$, then there exists an injection from $A$ into $B$ if and only if there exists an injection from $A'$ into $B'$.

Proof

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Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $3$: Cardinality: Exercise $3$