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 ap : set set set axiom ap_Pi: !x:set.!f:set set.!y:set.!z:set.y iIn Pi x f -> z iIn x -> ap y z iIn f z const Sep : set (set prop) set axiom SepI: !x:set.!p:set prop.!y:set.y iIn x -> p y -> y iIn Sep x p const Repl : set (set set) set axiom ReplE_impred: !x:set.!f:set set.!y:set.y iIn Repl x f -> !P:prop.(!z:set.z iIn x -> y = f z -> P) -> P const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z const SNoS_ : set set const SNoLev : set set 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 axiom SepE: !x:set.!p:set prop.!y:set.y iIn Sep x p -> y iIn x & p y const SNoCut : set set set axiom SNoCut_Le: !x:set.!y:set.!z:set.!w:set.SNoCutP x y -> SNoCutP z w -> (!u:set.u iIn x -> u < SNoCut z w) -> (!u:set.u iIn w -> SNoCut x y < u) -> SNoCut x y <= SNoCut z w const real : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set const minus_SNo : set set term - = minus_SNo axiom SNo_approx_real_rep: !x:set.x iIn real -> !P:prop.(!y:set.y iIn setexp (SNoS_ omega) omega -> !z:set.z iIn setexp (SNoS_ omega) omega -> (!w:set.w iIn omega -> ap y w < x) -> (!w:set.w iIn omega -> x < ap y w + eps_ w) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap y u < ap y w) -> (!w:set.w iIn omega -> (ap z w + - eps_ w) < x) -> (!w:set.w iIn omega -> x < ap z w) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap z w < ap z u) -> SNoCutP (Repl omega (ap y)) (Repl omega (ap z)) -> x = SNoCut (Repl omega (ap y)) (Repl omega (ap z)) -> P) -> P const ordsucc : set set const binunion : set set set const famunion : set (set set) set const SNoEq_ : set set set prop axiom SNoCutP_SNoCut_impred: !x:set.!y:set.SNoCutP x y -> !P:prop.(SNo (SNoCut x y) -> SNoLev (SNoCut x y) iIn ordsucc (binunion (famunion x \z:set.ordsucc (SNoLev z)) (famunion y \z:set.ordsucc (SNoLev z))) -> (!z:set.z iIn x -> z < SNoCut x y) -> (!z:set.z iIn y -> SNoCut x y < z) -> (!z:set.SNo z -> (!w:set.w iIn x -> w < z) -> (!w:set.w iIn y -> z < w) -> Subq (SNoLev (SNoCut x y)) (SNoLev z) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) z) -> P) -> P lemma !x:set.!y:set.x iIn setexp real omega -> y iIn setexp real omega -> ~ (?z:set.z iIn real & !w:set.w iIn omega -> ap x w <= z & z <= ap y w) -> (!z:set.z iIn omega -> SNo (ap x z)) -> (!z:set.z iIn omega -> SNo (ap y z)) -> (!z:set.z iIn omega -> !w:set.w iIn omega -> ap x z <= ap y w) -> Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (SNoS_ omega) -> Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) (SNoS_ omega) -> SNoCutP (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) -> SNo (SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z)) -> (!z:set.z iIn Sep (SNoS_ omega) (\w:set.?u:set.u iIn omega & w < ap x u) -> z < 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)) -> (!z:set.z iIn Sep (SNoS_ omega) (\w:set.?u:set.u iIn omega & ap y u < w) -> 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) < z) -> (!z:set.SNo z -> (!w:set.w iIn Sep (SNoS_ omega) (\u:set.?v:set.v iIn omega & u < ap x v) -> w < z) -> (!w:set.w iIn Sep (SNoS_ omega) (\u:set.?v:set.v iIn omega & ap y v < u) -> z < w) -> Subq (SNoLev (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))) (SNoLev z) & SNoEq_ (SNoLev (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))) (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)) z) -> ~ !z:set.z iIn omega -> ap x z <= 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) var x:set var y:set hyp x iIn setexp real omega hyp y iIn setexp real omega hyp ~ ?z:set.z iIn real & !w:set.w iIn omega -> ap x w <= z & z <= ap y w hyp !z:set.z iIn omega -> SNo (ap x z) hyp !z:set.z iIn omega -> SNo (ap y z) hyp !z:set.z iIn omega -> !w:set.w iIn omega -> ap x z <= ap y w hyp Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (SNoS_ omega) hyp Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) (SNoS_ omega) claim ~ SNoCutP (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z)