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



Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ?
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.


