Basic set-theoretic axioms inherited by a transitive class
= Basic set-theoretic axioms inherited by a transitive class
If a transitive class $M$ contains the empty set and is closed under pairing and union, then $(M,\in)$ satisfies extensionality, empty set, pairing, and union. Transitivity makes all members of each $x\in M$ visible inside $M$, so the ambient witnesses have the same required membership relations internally.