Ordinal analysis of non-monotone -definable inductive definitions

Exploiting the fact that Π 1 0 -definable non-monotone inductive definitions have the same closure ordinal as arbitrary arithmetically definable monotone inductive definitions, we show that the proof theoretic ordinal of an axiomatization ( Π 1 0 - FXP ) 0  of Π 1 0 -definable non-monotone inductive definitions coincides with the proof theoretic ordinal of the theory ID 1  of arithmetically definable monotone inductive definitions.