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 const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w term nIn = \x:set.\y:set.~ x iIn y const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const ordsucc : set set axiom omega_ordsucc: !x:set.x iIn omega -> ordsucc x iIn omega const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom ordinal_ordsucc: !x:set.ordinal x -> ordinal (ordsucc x) axiom ordsuccI2: !x:set.x iIn ordsucc x axiom In_irref: !x:set.nIn x x axiom FalseE: ~ False axiom ordinal_In_Or_Subq: !x:set.!y:set.ordinal x -> ordinal y -> x iIn y | Subq y x axiom omega_TransSet: TransSet omega const SNo_ : set set prop axiom SNoLev_: !x:set.SNo x -> SNo_ (SNoLev x) x const SNoS_ : set set axiom SNoS_I: !x:set.ordinal x -> !y:set.!z:set.z iIn x -> SNo_ z y -> y iIn SNoS_ x const real : set axiom SNoS_omega_real: Subq (SNoS_ omega) real const Sep : set (set prop) set const ap : set set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const SNoCut : set set set const SNoEq_ : set set set prop var x:set var y:set var z:set hyp ~ ?w:set.w iIn real & !u:set.u iIn omega -> ap x u <= w & w <= ap y u hyp !w:set.w iIn omega -> SNo (ap y w) hyp SNo (SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w)) hyp !w:set.SNo w -> (!u:set.u iIn Sep (SNoS_ omega) (\v:set.?x2:set.x2 iIn omega & v < ap x x2) -> u < w) -> (!u:set.u iIn Sep (SNoS_ omega) (\v:set.?x2:set.x2 iIn omega & ap y x2 < v) -> w < u) -> Subq (SNoLev (SNoCut (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & u < ap x v) (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & ap y v < u))) (SNoLev w) & SNoEq_ (SNoLev (SNoCut (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & u < ap x v) (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & ap y v < u))) (SNoCut (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & u < ap x v) (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & ap y v < u)) w hyp !w:set.w iIn omega -> ap x w <= SNoCut (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & u < ap x v) (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & ap y v < u) & SNoCut (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & u < ap x v) (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & ap y v < u) <= ap y w hyp z iIn SNoS_ omega hyp ~ (z iIn Sep (SNoS_ omega) (\w:set.?u:set.u iIn omega & w < ap x u) | z iIn Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w) hyp SNoLev z iIn omega hyp ordinal (SNoLev z) hyp SNo z hyp !w:set.w iIn Sep (SNoS_ omega) (\u:set.?v:set.v iIn omega & u < ap x v) -> w < z claim ~ !w:set.w iIn Sep (SNoS_ omega) (\u:set.?v:set.v iIn omega & ap y v < u) -> z < w