const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y const In_rec_ii : (set (set set set) set set) set set set const If_ii : prop (set set) (set set) set set const If_i : prop set set set const SNoS_ : set set const ordsucc : set set const SNoLev : set set const Eps_i : (set prop) set term SNo_rec_i = \P:set (set set) set.\x:set.In_rec_ii (\y:set.\g:set set set.If_ii (ordinal y) (\z:set.If_i (z iIn SNoS_ (ordsucc y)) (P z \w:set.g (SNoLev w) w) (Eps_i \w:set.True)) \z:set.Eps_i \w:set.True) (SNoLev x) x term nIn = \x:set.\y:set.~ x iIn y const SNo : set prop axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom In_rec_ii_eq: !P:set (set set set) set set.(!x:set.!g:set set set.!h:set set set.(!y:set.y iIn x -> g y = h y) -> P x g = P x h) -> !x:set.In_rec_ii P x = P x (In_rec_ii P) lemma !P:set (set set) set.!x:set.SNo x -> ordinal (SNoLev x) -> If_ii (ordinal (SNoLev x)) (\y:set.If_i (y iIn SNoS_ (ordsucc (SNoLev x))) (P y \z:set.In_rec_ii (\w:set.\g:set set set.If_ii (ordinal w) (\u:set.If_i (u iIn SNoS_ (ordsucc w)) (P u \v:set.g (SNoLev v) v) (Eps_i \v:set.True)) \u:set.Eps_i \v:set.True) (SNoLev z) z) (Eps_i \z:set.True)) (\y:set.Eps_i \z:set.True) x = P x \y:set.In_rec_ii (\z:set.\g:set set set.If_ii (ordinal z) (\w:set.If_i (w iIn SNoS_ (ordsucc z)) (P w \u:set.g (SNoLev u) u) (Eps_i \u:set.True)) \w:set.Eps_i \u:set.True) (SNoLev y) y var P:set (set set) set var x:set hyp !y:set.SNo y -> !f:set set.!f2:set set.(!z:set.z iIn SNoS_ (SNoLev y) -> f z = f2 z) -> P y f = P y f2 hyp SNo x claim (!y:set.!g:set set set.!h:set set set.(!z:set.z iIn y -> g z = h z) -> If_ii (ordinal y) (\z:set.If_i (z iIn SNoS_ (ordsucc y)) (P z \w:set.g (SNoLev w) w) (Eps_i \w:set.True)) (\z:set.Eps_i \w:set.True) = If_ii (ordinal y) (\z:set.If_i (z iIn SNoS_ (ordsucc y)) (P z \w:set.h (SNoLev w) w) (Eps_i \w:set.True)) \z:set.Eps_i \w:set.True) -> In_rec_ii (\y:set.\g:set set set.If_ii (ordinal y) (\z:set.If_i (z iIn SNoS_ (ordsucc y)) (P z \w:set.g (SNoLev w) w) (Eps_i \w:set.True)) \z:set.Eps_i \w:set.True) (SNoLev x) x = P x \y:set.In_rec_ii (\z:set.\g:set set set.If_ii (ordinal z) (\w:set.If_i (w iIn SNoS_ (ordsucc z)) (P w \u:set.g (SNoLev u) u) (Eps_i \u:set.True)) \w:set.Eps_i \u:set.True) (SNoLev y) y