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 In_irref: !x:set.nIn x x const SNoLev : set set const SNoCut : set set set const SNoEq_ : set set set prop var x:set var y:set var z:set var w:set hyp !u:set.u iIn w -> SNoCut x y < u hyp !u:set.u iIn z -> SNo u hyp SNo (SNoCut x y) hyp SNo (SNoCut z w) hyp !u:set.u iIn z -> u < SNoCut z w hyp !u:set.SNo u -> (!v:set.v iIn z -> v < u) -> (!v:set.v iIn w -> u < v) -> Subq (SNoLev (SNoCut z w)) (SNoLev u) & SNoEq_ (SNoLev (SNoCut z w)) (SNoCut z w) u hyp SNoCut z w < SNoCut x y hyp SNoLev (SNoCut x y) iIn SNoLev (SNoCut z w) claim ~ !u:set.u iIn z -> u < SNoCut x y