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 const ordinal : set prop const ordsucc : set set axiom ordinal_ordsucc: !x:set.ordinal x -> ordinal (ordsucc x) const SNoCutP : set set prop const SNoLev : set set const binunion : set set set const famunion : set (set set) set const SNoCut : set set set lemma !x:set.!y:set.!z:set.ordinal x -> (!w:set.w iIn x -> ordsucc w iIn x) -> SNoCutP y z -> (!w:set.w iIn y -> SNoLev w iIn x) -> (!w:set.w iIn z -> SNoLev w iIn x) -> ordinal (binunion (famunion y \w:set.ordsucc (SNoLev w)) (famunion z \w:set.ordsucc (SNoLev w))) -> ordinal (ordsucc x) -> SNoLev (SNoCut y z) iIn ordsucc x const SNoS_ : set set var x:set var y:set var z:set hyp ordinal x hyp !w:set.w iIn x -> ordsucc w iIn x hyp Subq y (SNoS_ x) hyp Subq z (SNoS_ x) hyp SNoCutP y z hyp !w:set.w iIn y -> SNoLev w iIn x hyp !w:set.w iIn z -> SNoLev w iIn x claim ordinal (binunion (famunion y \w:set.ordsucc (SNoLev w)) (famunion z \w:set.ordsucc (SNoLev w))) -> SNoLev (SNoCut y z) iIn ordsucc x