The Regular Extension Axiom (REA) is a foundational axiom which asserts the existence of arbitrarily large regular cardinal-like sets. It has several variants, some of which are provable in ZF, some of which are provable from the axiom of choice or weaker variants thereof such as SVC, and some of which are not even provable in ZFC. REA is usually considered in the context of CZF.
There is some discussion here.
uREA is REA for union closed regular sets. In CZF it implies the set generated axiom (SGA):
For each set S and each subset Z of Fin(S) × Pow(Pow(S)), the class
M(Z) = {α ∈ Pow(S) | ∀(σ, Γ) ∈ Z[σ ⊆ α ⇒ ∃U ∈ Γ(U ⊆ α)]}
is set-generated. This axiom is also implies by relativised dependent choice?.
Michael Rathjen? and Robert Lubarsky?, On the regular extension axiom and its variants, PDF
Peter Aczel, Hajime Ishihara, Takako Nemoto and Yasushi Sangu, Generalized geometric theories and set-generated classes, PDF