const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term nIn = \x:set.\y:set.~ x iIn y 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 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const SNoCut : set set set const SNoLev : set set const SNoEq_ : set set set prop const binintersect : set set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.(!v:set.v iIn y -> SNo v) -> SNo (SNoCut x y) -> (!v:set.v iIn y -> SNoCut x y < v) -> (!v:set.SNo v -> (!x2:set.x2 iIn x -> x2 < v) -> (!x2:set.x2 iIn y -> v < x2) -> Subq (SNoLev (SNoCut x y)) (SNoLev v) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) v) -> SNo u -> SNoLev u iIn binintersect (SNoLev (SNoCut z w)) (SNoLev (SNoCut x y)) -> u < SNoCut x y -> (!v:set.v iIn x -> v < u) -> ~ !v:set.v iIn y -> u < v var x:set var y:set var z:set var w:set var u:set hyp !v:set.v iIn x -> v < SNoCut z w hyp !v:set.v iIn x -> SNo v hyp !v:set.v iIn y -> SNo v hyp SNo (SNoCut x y) hyp !v:set.v iIn y -> SNoCut x y < v hyp !v:set.SNo v -> (!x2:set.x2 iIn x -> x2 < v) -> (!x2:set.x2 iIn y -> v < x2) -> Subq (SNoLev (SNoCut x y)) (SNoLev v) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) v hyp SNo (SNoCut z w) hyp SNo u hyp SNoLev u iIn binintersect (SNoLev (SNoCut z w)) (SNoLev (SNoCut x y)) hyp SNoCut z w < u hyp u < SNoCut x y claim ~ !v:set.v iIn x -> v < u