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 SNoS_ : set set const omega : set const SNoLev : set set const SNoCut : set set set const ordsucc : set set axiom SNoCutP_SNoCut_omega: !x:set.Subq x (SNoS_ omega) -> !y:set.Subq y (SNoS_ omega) -> SNoCutP x y -> SNoLev (SNoCut x y) iIn ordsucc omega const Repl : set (set set) set const ap : set set set lemma !x:set.!y:set.!P:prop.(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) -> SNoCutP (Repl omega (ap x)) (Repl omega (ap y)) -> SNo (SNoCut (Repl omega (ap x)) (Repl omega (ap y))) -> (!z:set.z iIn Repl omega (ap x) -> z < SNoCut (Repl omega (ap x)) (Repl omega (ap y))) -> (!z:set.z iIn Repl omega (ap y) -> SNoCut (Repl omega (ap x)) (Repl omega (ap y)) < z) -> Subq (Repl omega (ap x)) (SNoS_ omega) -> Subq (Repl omega (ap y)) (SNoS_ omega) -> SNoLev (SNoCut (Repl omega (ap x)) (Repl omega (ap y))) iIn ordsucc omega -> P var x:set var y:set var P:prop hyp y iIn setexp (SNoS_ omega) omega 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 SNoCutP (Repl omega (ap x)) (Repl omega (ap y)) hyp SNo (SNoCut (Repl omega (ap x)) (Repl omega (ap y))) hyp !z:set.z iIn Repl omega (ap x) -> z < SNoCut (Repl omega (ap x)) (Repl omega (ap y)) hyp !z:set.z iIn Repl omega (ap y) -> SNoCut (Repl omega (ap x)) (Repl omega (ap y)) < z hyp Subq (Repl omega (ap x)) (SNoS_ omega) claim Subq (Repl omega (ap y)) (SNoS_ omega) -> P