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 SNoS_SNoLev: !x:set.SNo x -> x iIn SNoS_ (ordsucc (SNoLev x)) axiom If_ii_1: !P:prop.!f:set set.!f2:set set.P -> If_ii P f f2 = f lemma !P:set (set set) set.!x:set.SNo x -> x iIn SNoS_ (ordsucc (SNoLev x)) -> If_i (x iIn SNoS_ (ordsucc (SNoLev 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) (Eps_i \y:set.True) = 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 SNo x claim 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