const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const SNoS_ : set set const SNoLev : set set 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 const ap : set set set const Sigma : set (set set) set axiom beta: !x:set.!f:set set.!y:set.y iIn x -> ap (Sigma x f) y = f y const nat_primrec : set (set set set) set set const Eps_i : (set prop) set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set const Empty : set const ordsucc : set set var x:set var y:set var z:set hyp !w:set.w iIn omega -> nat_primrec (Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ Empty) (\u:set.\v:set.Eps_i \x2:set.x2 iIn SNoS_ omega & x2 < x & x < x2 + eps_ (ordsucc u) & v < x2) w iIn SNoS_ omega & ap (Sigma omega (nat_primrec (Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ Empty) \u:set.\v:set.Eps_i \x2:set.x2 iIn SNoS_ omega & x2 < x & x < x2 + eps_ (ordsucc u) & v < x2)) w < x & x < ap (Sigma omega (nat_primrec (Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ Empty) \u:set.\v:set.Eps_i \x2:set.x2 iIn SNoS_ omega & x2 < x & x < x2 + eps_ (ordsucc u) & v < x2)) w + eps_ w & !u:set.u iIn w -> SNo (ap (Sigma omega (nat_primrec (Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ Empty) \v:set.\x2:set.Eps_i \y2:set.y2 iIn SNoS_ omega & y2 < x & x < y2 + eps_ (ordsucc v) & x2 < y2)) u) -> ap (Sigma omega (nat_primrec (Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ Empty) \v:set.\x2:set.Eps_i \y2:set.y2 iIn SNoS_ omega & y2 < x & x < y2 + eps_ (ordsucc v) & x2 < y2)) u < ap (Sigma omega (nat_primrec (Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ Empty) \v:set.\x2:set.Eps_i \y2:set.y2 iIn SNoS_ omega & y2 < x & x < y2 + eps_ (ordsucc v) & x2 < y2)) w hyp y iIn omega hyp z iIn y claim z iIn omega -> SNo (ap (Sigma omega (nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) \w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v)) z)