A theorem stating that if there exists an injective function from A into B and an injective function from B into A, then there exists a bijection between A and B; mutual embeddability implies equipotence.

Principle

Mutual embeddings can be combined into a bijection by decomposing elements into chains or orbits under the compositions of injections and carefully defining a pairing on those components; equality of cardinalities follows from mutual injections without invoking choice.

Demonstration

To show Z and 2Z (integers and even integers) have the same cardinality, provide injections Z → 2Z (n ↦ 2n) and 2Z → Z (identity inclusion); Cantor–Bernstein–Schroeder then guarantees a bijection between Z and 2Z even though a simple explicit bijection can also be given.

Misapplication

Assuming that the existence of injections in just one direction suffices for a bijection, or trying to apply the theorem to proper classes without verifying set-theoretic hypotheses; misusing it to claim canonicity of the bijection rather than mere existence.

Consequence

Provides a robust tool for comparing cardinalities: it makes the relation ‘there exists an injection’ into a partial order whose antisymmetry is witnessed by bijections; it is fundamental in cardinal arithmetic and combinatorial set theory.

Reversal

If injections do not exist in both directions, Cantor–Bernstein–Schroeder gives no conclusion; mutual non-embeddability does not imply strict inequality of cardinalities without further structure or proof.

Boundary

Requires genuine injections between sets; it does not construct a canonical bijection in general and it applies to sets (not arbitrary classes) within standard set theory frameworks.

Semantic Tension

Often weighed against Cantor's theorem: CB–S can show equality from mutual injections while Cantor's theorem ensures strict inequalities for power sets; the two coexist without contradiction but operate on different phenomena (mutual embedding vs diagonal non-surjectivity).

Synthesis

Cantor–Bernstein–Schroeder converts mutual injective embeddability into a bijection, allowing cardinal equality to be established from two one-sided embeddings and cementing mutual embeddability as the correct notion of same cardinality in set theory.