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 omega : set const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const Empty : set axiom SNo_0: SNo Empty axiom SNo_eps_pos: !x:set.x iIn omega -> Empty < eps_ x axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const Sing : set set axiom SingE: !x:set.!y:set.y iIn Sing x -> y = x axiom In_no2cycle: !x:set.!y:set.x iIn y -> ~ y iIn x axiom FalseE: ~ False const SNoLev : set set const ordsucc : set set axiom SNoLev_eps_: !x:set.x iIn omega -> SNoLev (eps_ x) = ordsucc x const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x axiom nat_ordsucc: !x:set.nat_p x -> nat_p (ordsucc x) const ordinal : set prop axiom nat_p_ordinal: !x:set.nat_p x -> ordinal x const SNo_ : set set prop axiom SNoLev_: !x:set.SNo x -> SNo_ (SNoLev x) x const SNoS_ : set set axiom SNoS_I: !x:set.ordinal x -> !y:set.!z:set.z iIn x -> SNo_ z y -> y iIn SNoS_ x axiom SingI: !x:set.x iIn Sing x const SNoEq_ : set set set prop axiom eps_SNo_eq: !x:set.nat_p x -> !y:set.y iIn SNoS_ (ordsucc x) -> Empty < y -> SNoEq_ (SNoLev y) (eps_ x) y -> ?z:set.z iIn x & y = eps_ z const binintersect : set set set axiom SNoLtE: !x:set.!y:set.SNo x -> SNo y -> x < y -> !P:prop.(!z:set.SNo z -> SNoLev z iIn binintersect (SNoLev x) (SNoLev y) -> SNoEq_ (SNoLev z) z x -> SNoEq_ (SNoLev z) z y -> x < z -> z < y -> nIn (SNoLev z) x -> SNoLev z iIn y -> P) -> (SNoLev x iIn SNoLev y -> SNoEq_ (SNoLev x) x y -> SNoLev x iIn y -> P) -> (SNoLev y iIn SNoLev x -> SNoEq_ (SNoLev y) x y -> nIn (SNoLev y) x -> P) -> P const Repl : set (set set) set const SNoCut : set set set lemma !x:set.!y:set.x iIn omega -> (!z:set.z iIn Repl x eps_ -> SNo z) -> SNo (SNoCut (Sing Empty) (Repl x eps_)) -> (!z:set.z iIn Repl x eps_ -> SNoCut (Sing Empty) (Repl x eps_) < z) -> (!z:set.SNo z -> (!w:set.w iIn Sing Empty -> w < z) -> (!w:set.w iIn Repl x eps_ -> z < w) -> Subq (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoLev z) & SNoEq_ (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoCut (Sing Empty) (Repl x eps_)) z) -> SNo y -> SNoLev y iIn binintersect (SNoLev (eps_ x)) (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) -> eps_ x < y -> y < SNoCut (Sing Empty) (Repl x eps_) -> ~ !z:set.z iIn Sing Empty -> z < y lemma !x:set.x iIn omega -> SNo (SNoCut (Sing Empty) (Repl x eps_)) -> (!y:set.y iIn Sing Empty -> y < SNoCut (Sing Empty) (Repl x eps_)) -> (!y:set.y iIn Repl x eps_ -> SNoCut (Sing Empty) (Repl x eps_) < y) -> SNoLev (SNoCut (Sing Empty) (Repl x eps_)) iIn ordsucc x -> SNoEq_ (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (eps_ x) (SNoCut (Sing Empty) (Repl x eps_)) -> ~ ?y:set.y iIn x & SNoCut (Sing Empty) (Repl x eps_) = eps_ y var x:set hyp x iIn omega hyp !y:set.y iIn Repl x eps_ -> SNo y 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 !y:set.SNo y -> (!z:set.z iIn Sing Empty -> z < y) -> (!z:set.z iIn Repl x eps_ -> y < z) -> Subq (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoLev y) & SNoEq_ (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoCut (Sing Empty) (Repl x eps_)) y hyp SNoLev (SNoCut (Sing Empty) (Repl x \y:set.eps_ y)) iIn ordsucc x claim ~ eps_ x < SNoCut (Sing Empty) (Repl x eps_)