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 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const Repl : set (set set) set const eps_ : set set const SNoCut : set set set const Sing : set set const Empty : set const SNoLev : set set const SNoEq_ : set set set prop const binintersect : set set set lemma !x:set.!y:set.(!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_))) -> y < SNoCut (Sing Empty) (Repl x eps_) -> (!z:set.z iIn Sing Empty -> z < y) -> ~ !z:set.z iIn Repl x eps_ -> y < z const omega : set var x:set var y:set hyp x iIn omega hyp !z:set.z iIn Repl x eps_ -> SNo z hyp SNo (SNoCut (Sing Empty) (Repl x eps_)) hyp !z:set.z iIn Repl x eps_ -> SNoCut (Sing Empty) (Repl x eps_) < z hyp !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 hyp SNo y hyp SNoLev y iIn binintersect (SNoLev (eps_ x)) (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) hyp eps_ x < y hyp y < SNoCut (Sing Empty) (Repl x eps_) claim ~ !z:set.z iIn Sing Empty -> z < y