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 axiom ordsuccE: !x:set.!y:set.y iIn ordsucc x -> y iIn x | y = x const SNo : set prop const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P lemma !x:set.!g:set set set.!h:set set set.!y:set.!z:set.(!w:set.w iIn x -> g w = h w) -> ordinal x -> SNoLev y iIn ordsucc x -> SNoLev z iIn SNoLev y -> SNoLev z iIn x -> g (SNoLev z) z = h (SNoLev z) z var P:set (set set) set var x:set var g:set set set var h:set set set var y:set hyp !z:set.SNo z -> !f:set set.!f2:set set.(!w:set.w iIn SNoS_ (SNoLev z) -> f w = f2 w) -> P z f = P z f2 hyp !z:set.z iIn x -> g z = h z hyp ordinal x hyp y iIn SNoS_ (ordsucc x) claim ordinal (ordsucc x) -> P y (\z:set.g (SNoLev z) z) = P y \z:set.h (SNoLev z) z