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 term nIn = \x:set.\y:set.~ x iIn y const Repl : set (set set) set axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f axiom SNoLt_irref: !x:set.~ x < x const SNoEq_ : set set set prop const SNoLev : set set const SNoCut : set set set const Sing : set set const Empty : set const eps_ : set set const ordsucc : set set const omega : set var x:set hyp x iIn omega hyp SNo (SNoCut (Sing Empty) (Repl x eps_)) hyp !y:set.y iIn Sing Empty -> y < SNoCut (Sing Empty) (Repl x eps_) hyp !y:set.y iIn Repl x eps_ -> SNoCut (Sing Empty) (Repl x eps_) < y hyp SNoLev (SNoCut (Sing Empty) (Repl x eps_)) iIn ordsucc x hyp SNoEq_ (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (eps_ x) (SNoCut (Sing Empty) (Repl x eps_)) claim ~ ?y:set.y iIn x & SNoCut (Sing Empty) (Repl x eps_) = eps_ y