The Vopenka principle is inequivalent to but conservative over the Vopenka scheme
Abstract
The Vopenka principle, which asserts that every proper class of first-order structures in a common language admits an elementary embedding between two of its members, is not equivalent over GBC to the first-order Vopenka scheme, which makes the Vopenka assertion only for the first-order definable classes of structures. Nevertheless, the two Vopenka axioms are equiconsistent and they have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vopenka principle is conservative over ZFC plus the Vopenka scheme for first-order assertions in the language of set theory.
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