const In : set set prop term iIn = In infix iIn 2000 2000 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 Subq = \x:set.\y:set.!z:set.z iIn x -> z 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 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 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 const Repl : set (set set) set const SNoCut : set set set const ordsucc : set set lemma !x:set.!y:set.!P:prop.x iIn setexp (SNoS_ omega) omega -> y iIn setexp (SNoS_ omega) omega -> (!z:set.z iIn omega -> !w:set.w iIn omega -> ap x z < ap y w) -> (SNoCutP (Repl omega (ap x)) (Repl omega (ap y)) -> SNo (SNoCut (Repl omega (ap x)) (Repl omega (ap y))) -> SNoLev (SNoCut (Repl omega (ap x)) (Repl omega (ap y))) iIn ordsucc omega -> SNoCut (Repl omega (ap x)) (Repl omega (ap y)) iIn SNoS_ (ordsucc omega) -> (!z:set.z iIn omega -> ap x z < SNoCut (Repl omega (ap x)) (Repl omega (ap y))) -> (!z:set.z iIn omega -> SNoCut (Repl omega (ap x)) (Repl omega (ap y)) < ap y z) -> P) -> (!z:set.z iIn omega -> SNo (ap x z)) -> (!z:set.z iIn omega -> SNo (ap y z)) -> P var x:set var y:set var P:prop hyp x iIn setexp (SNoS_ omega) omega hyp y iIn setexp (SNoS_ omega) omega hyp !z:set.z iIn omega -> !w:set.w iIn omega -> ap x z < ap y w hyp SNoCutP (Repl omega (ap x)) (Repl omega (ap y)) -> SNo (SNoCut (Repl omega (ap x)) (Repl omega (ap y))) -> SNoLev (SNoCut (Repl omega (ap x)) (Repl omega (ap y))) iIn ordsucc omega -> SNoCut (Repl omega (ap x)) (Repl omega (ap y)) iIn SNoS_ (ordsucc omega) -> (!z:set.z iIn omega -> ap x z < SNoCut (Repl omega (ap x)) (Repl omega (ap y))) -> (!z:set.z iIn omega -> SNoCut (Repl omega (ap x)) (Repl omega (ap y)) < ap y z) -> P claim (!z:set.z iIn omega -> SNo (ap x z)) -> P