const SNo : set prop const ordinal : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const ordsucc : set set lemma !P:set set set prop.!x:set.!y:set.!z:set.(!w:set.ordinal w -> !u:set.ordinal u -> !v:set.ordinal v -> !x2:set.x2 iIn SNoS_ w -> !y2:set.y2 iIn SNoS_ u -> !z2:set.z2 iIn SNoS_ v -> P x2 y2 z2) -> SNo y -> SNo z -> ordinal (ordsucc (SNoLev x)) -> x iIn SNoS_ (ordsucc (SNoLev x)) -> ordinal (SNoLev y) -> P x y z var P:set set set prop var x:set var y:set var z:set hyp !w:set.ordinal w -> !u:set.ordinal u -> !v:set.ordinal v -> !x2:set.x2 iIn SNoS_ w -> !y2:set.y2 iIn SNoS_ u -> !z2:set.z2 iIn SNoS_ v -> P x2 y2 z2 hyp SNo x hyp SNo y hyp SNo z hyp ordinal (ordsucc (SNoLev x)) claim x iIn SNoS_ (ordsucc (SNoLev x)) -> P x y z