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 ordsucc : set set axiom omega_ordsucc: !x:set.x iIn omega -> ordsucc x iIn omega const SNo : set prop const SNo_ : set set prop const SNoLev : set set const SNo_extend1 : set set axiom SNo_extend1_SNo_: !x:set.SNo x -> SNo_ (ordsucc (SNoLev x)) (SNo_extend1 x) 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 const binintersect : set set set const SNoElts_ : set set axiom SNo_extend1_restr_eq: !x:set.SNo x -> x = binintersect (SNo_extend1 x) (SNoElts_ (SNoLev x)) var x:set var f:set set var f2:set set var y:set var z:set hyp nat_p x hyp !w:set.x iIn w -> f2 w = f (binintersect w (SNoElts_ x)) hyp f z = y hyp SNoLev z = x hyp SNo z hyp SNoLev (SNo_extend1 z) = ordsucc x claim x iIn SNo_extend1 z -> ?w:set.w iIn Sep (SNoS_ omega) (\u:set.SNoLev u = ordsucc x) & f2 w = y