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 PNoEq_ : set (set prop) (set prop) prop term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z const SNo : set prop const ordinal : set prop axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom or3I1: !P:prop.!Q:prop.!R:prop.P -> P | Q | R axiom SNo_eq: !x:set.!y:set.SNo x -> SNo y -> SNoLev x = SNoLev y -> SNoEq_ (SNoLev x) x y -> x = y axiom or3I2: !P:prop.!Q:prop.!R:prop.Q -> P | Q | R axiom or3I3: !P:prop.!Q:prop.!R:prop.R -> P | Q | R axiom PNoLt_trichotomy_or: !x:set.!y:set.!p:set prop.!q:set prop.ordinal x -> ordinal y -> PNoLt x p y q | x = y & PNoEq_ x p q | PNoLt y q x p claim !x:set.!y:set.SNo x -> SNo y -> x < y | x = y | y < x