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 TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y term nIn = \x:set.\y:set.~ x iIn y term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z axiom iffI: !P:prop.!Q:prop.(P -> Q) -> (Q -> P) -> (P <-> Q) const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLtI3: !x:set.!y:set.SNoLev y iIn SNoLev x -> SNoEq_ (SNoLev y) x y -> nIn (SNoLev y) x -> x < y axiom ordinal_In_SNoLt: !x:set.ordinal x -> !y:set.y iIn x -> y < x const SNo : set prop axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z var x:set var y:set var z:set hyp ordinal x hyp TransSet x hyp SNo y hyp SNo x hyp SNoLev y = x hyp ordinal z hyp !w:set.w iIn z -> w iIn x -> w iIn y hyp z iIn x hyp ~ z iIn y hyp SNo z claim SNoLev z = z -> y < x