# Is this set theory equivalent to ZFC?

Consider a variant of set theory with these axioms:

• Extensionality,
• Regularity (foundation),
• Separation,
• Powerset,
• Axiom of Choice, and
• Transitive closure of a set-like relation is set-like.

Note that it does not explicitly postulate Pairing, Union, Infinity and Replacement.

Question: Is this set theory equivalent to $$mathrm{ZFC}$$?