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