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 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 SNoS_ : set set const omega : set const ap : set set set const SNoCut : set set set const SNoLev : 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)) -> SNoCutP (Repl omega (ap x)) (Repl omega (ap y)) -> 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 hyp !z:set.z iIn omega -> SNo (ap x z) claim (!z:set.z iIn omega -> SNo (ap y z)) -> P