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 Empty : set axiom EmptyE: !x:set.nIn x Empty axiom FalseE: ~ False 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 axiom Empty_eq: !x:set.(!y:set.nIn y x) -> x = Empty const nat_p : set prop const ordsucc : set set axiom nat_0_in_ordsucc: !x:set.nat_p x -> Empty iIn ordsucc x axiom cases_1: !x:set.x iIn ordsucc Empty -> !p:set prop.p Empty -> p x const binunion : set set set axiom Subq_binunion_eq: !x:set.!y:set.Subq x y = (binunion x y = y) axiom xm: !P:prop.P | ~ P const omega : set const Sing : set set const eps_ : set set const famunion : set (set set) set const SNoLev : set set const SNoCut : set set set lemma !x:set.x iIn omega -> nat_p x -> SNoCutP (Sing Empty) (Repl x eps_) -> (!y:set.y iIn Repl x eps_ -> SNo y) -> famunion (Sing Empty) (\y:set.ordsucc (SNoLev y)) = ordsucc Empty -> (x != Empty -> famunion (Repl x eps_) (\y:set.ordsucc (SNoLev y)) = ordsucc x) -> binunion (famunion (Sing Empty) \y:set.ordsucc (SNoLev y)) (famunion (Repl x eps_) \y:set.ordsucc (SNoLev y)) = ordsucc x -> eps_ x = SNoCut (Sing Empty) (Repl x eps_) lemma !x:set.famunion (Sing Empty) (\y:set.ordsucc (SNoLev y)) = ordsucc Empty -> x = Empty -> Repl x eps_ = Empty -> binunion (famunion (Sing Empty) \y:set.ordsucc (SNoLev y)) (famunion (Repl x eps_) \y:set.ordsucc (SNoLev y)) = ordsucc x var x:set hyp x iIn omega hyp nat_p x hyp SNoCutP (Sing Empty) (Repl x eps_) hyp !y:set.y iIn Repl x eps_ -> SNo y hyp famunion (Sing Empty) (\y:set.ordsucc (SNoLev y)) = ordsucc Empty claim (x != Empty -> famunion (Repl x eps_) (\y:set.ordsucc (SNoLev y)) = ordsucc x) -> eps_ x = SNoCut (Sing Empty) (Repl x eps_)