const bij : set set (set set) prop term equip = \x:set.\y:set.?f:set set.bij x y f const In : set set prop term iIn = In infix iIn 2000 2000 term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) term nIn = \x:set.\y:set.~ x iIn y term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const nat_p : set prop axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const SNo : set prop const SNoLev : set set const SNo_ : set set prop const binintersect : set set set const SNoElts_ : set set axiom restr_SNo_: !x:set.SNo x -> !y:set.y iIn SNoLev x -> SNo_ y (binintersect x (SNoElts_ y)) const SNoS_ : set set axiom SNoS_I: !x:set.ordinal x -> !y:set.!z:set.z iIn x -> SNo_ z y -> y iIn SNoS_ x const Sep : set (set prop) set axiom SepI: !x:set.!p:set prop.!y:set.y iIn x -> p y -> y iIn Sep x p axiom restr_SNo: !x:set.SNo x -> !y:set.y iIn SNoLev x -> SNo (binintersect x (SNoElts_ y)) const ordsucc : set set var x:set var y:set var P:prop hyp nat_p x hyp SNoLev y = ordsucc x hyp SNo y hyp SNo y -> SNoLev y = ordsucc x -> binintersect y (SNoElts_ x) iIn Sep (SNoS_ omega) (\z:set.SNoLev z = x) -> SNo (binintersect y (SNoElts_ x)) -> SNoLev (binintersect y (SNoElts_ x)) = x -> P hyp x iIn SNoLev y claim SNoLev (binintersect y (SNoElts_ x)) = x -> P