In order to define a specific Schröder–Bernstein property one should decide: In the classical (Cantor–)Schröder–Bernstein theorem: Not all statements of this form are true.
A Schröder–Bernstein property is a joint property of: Instead of the relation "be a part of" one may use a binary relation "be embeddable into" (embeddability) interpreted as "be similar to some part of".
Also, embeddability is usually a preorder, and similarity is usually an equivalence relation (which is natural, but not provable in the absence of formal definitions).
The Schröder–Bernstein theorem for measurable spaces[2] states the Schröder–Bernstein property for the following case: In the Schröder–Bernstein theorem for operator algebras:[3] Taking into account that commutative von Neumann algebras are closely related to measurable spaces,[4] one may say that the Schröder–Bernstein theorem for operator algebras is in some sense a noncommutative counterpart of the Schröder–Bernstein theorem for measurable spaces.
are discussed by informal groups of mathematicians (see External Links below).