Chapter 15 of the third edition of Jech's textbook on set theory gives Vopenka's theorem as saying that if $V=L[A]$ where $A$ is a set of ordinals then $V$ is a set generic extension of $HOD$, whereas an article of Hugh Woodin gives Vopenka's theorem as saying (provably in $\textsf{ZF}$ alone) that for all ordinals $\alpha$, $HOD(V_{\alpha})$ is a symmetric generic extension of $HOD$, which seems at least at first sight to be a stronger statement. Is there a reference for the proof of the second version of the theorem?
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3$\begingroup$ The first place I'd look for this (If I had it handy) is "Intermediate submodels and generic extensions in set theory" Ann. Math. 101 (1975) 447--490. It may also be in the book "Theory of Semisets" by Vopenka and Hajek. $\endgroup$– Andreas BlassCommented Apr 30 at 15:21
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3$\begingroup$ What about Theorem 6 in The HOD Dichotomy $\endgroup$– Mohammad GolshaniCommented May 3 at 6:55
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1$\begingroup$ As a (tardy) addendum to Andreas Blass's comment: see Theorem C of Grigorieff's paper, availble via: https://%20https://people.math.wisc.edu/~awmille1/old/m873-03/grigor.pdf $\endgroup$– Ali EnayatCommented May 11 at 9:05
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