I have a proof in ZFC, using AC and the axiom of foundation, that given any proper class A, every set can be injected into A. I wonder if we could have a proof of this that does not use Foundation. Gérard Lang
No. This principle is known as The Injection Principle
See in Jech Axiom of Choice, Chapter 9, Problems 3 and 4 both give us a models of ZF+Atoms (so foundation fails) with choice in which the injection principle fails.