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The Vop\v{e}nka principle, which asserts that every proper class offirst-order structures in a common language admits an elementary embeddingbetween two of its members, is not equivalent over GBC to the first-orderVop\v{e}nka scheme, which makes the Vop\v{e}nka assertion only for thefirst-order definable classes of structures. Nevertheless, the two Vop\v{e}nkaaxioms are equiconsistent and they have exactly the same first-orderconsequences in the language of set theory. Specifically, GBC plus theVop\v{e}nka principle is conservative over ZFC plus the Vop\v{e}nka scheme forfirst-order assertions in the language of set theory.

The Vop\v{e}nka principle is inequivalent to but conservative over the Vop\v{e}nka scheme