Theorem 1.8.2.9 (Schröder-Bernstein).  Let κ, μ be cardinal numbers. Then the following are equivalent:
  1. κ ≤ μ and μ ≤ κ
  2. Iso(κ, μ) is nonempty
  3. κ = μ
No proof.
Remarks.

This is essentially the Schröder-Bernstein theorem together with the HLM-specific equality criterion on structures.

References.