Isn't Lawvere set theory equiconsistent with bounded Zermelo set theory? ZFC, its subtheories, and its extensions, serve as a universal yardstick for consisten

Isn't Lawvere set theory equiconsistent with bounded Zermelo set theory? ZFC, its subtheories, and its extensions, serve as a universal yardstick for consistency results. It's not really clear why that is the case, but it has so far proven to be so.