const PNoLt : set (set prop) set (set prop) prop const SNoLev : set set const In : set set prop term iIn = In infix iIn 2000 2000 term SNoLt = \x:set.\y:set.PNoLt (SNoLev x) (\z:set.z iIn x) (SNoLev y) \z:set.z iIn y term < = SNoLt infix < 2020 2020 const PNoLe : set (set prop) set (set prop) prop term SNoLe = \x:set.\y:set.PNoLe (SNoLev x) (\z:set.z iIn x) (SNoLev y) \z:set.z iIn y term <= = SNoLe infix <= 2020 2020 const SNo : set prop const ordinal : set prop axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom PNoLtLe_tra: !x:set.!y:set.!z:set.ordinal x -> ordinal y -> ordinal z -> !p:set prop.!q:set prop.!p2:set prop.PNoLt x p y q -> PNoLe y q z p2 -> PNoLt x p z p2 claim !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z