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 ordinal : set prop const PNoEq_ : set (set prop) (set prop) prop const PSNo : set (set prop) set axiom PNoEq_PSNo: !x:set.ordinal x -> !p:set prop.PNoEq_ x (\y:set.y iIn PSNo x p) p axiom PNoEq_sym_: !x:set.!p:set prop.!q:set prop.PNoEq_ x p q -> PNoEq_ x q p axiom PNoLtEq_tra: !x:set.!y:set.ordinal x -> ordinal y -> !p:set prop.!q:set prop.!p2:set prop.PNoLt x p y q -> PNoEq_ y q p2 -> PNoLt x p y p2 axiom PNoEqLt_tra: !x:set.!y:set.ordinal x -> ordinal y -> !p:set prop.!q:set prop.!p2:set prop.PNoEq_ x p q -> PNoLt x q y p2 -> PNoLt x p y p2 axiom SNoLev_PSNo: !x:set.ordinal x -> !p:set prop.SNoLev (PSNo x p) = x claim !x:set.!y:set.!p:set prop.!q:set prop.ordinal x -> ordinal y -> PNoLt x p y q -> PSNo x p < PSNo y q