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 SNoCutP : set set prop const SNo : set prop const SNoCut : set set set const SNoLev : set set const ordsucc : set set const binunion : set set set const famunion : set (set set) set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoEq_ : set set set prop axiom SNoCutP_SNoCut_impred: !x:set.!y:set.SNoCutP x y -> !P:prop.(SNo (SNoCut x y) -> SNoLev (SNoCut x y) iIn ordsucc (binunion (famunion x \z:set.ordsucc (SNoLev z)) (famunion y \z:set.ordsucc (SNoLev z))) -> (!z:set.z iIn x -> z < SNoCut x y) -> (!z:set.z iIn y -> SNoCut x y < z) -> (!z:set.SNo z -> (!w:set.w iIn x -> w < z) -> (!w:set.w iIn y -> z < w) -> Subq (SNoLev (SNoCut x y)) (SNoLev z) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) z) -> P) -> P const ordinal : set prop var x:set var y:set var z:set hyp !w:set.w iIn x -> ordsucc w iIn 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 hyp ordinal (binunion (famunion y \w:set.ordsucc (SNoLev w)) (famunion z \w:set.ordsucc (SNoLev w))) hyp ordinal (ordsucc x) claim Subq (ordsucc (binunion (famunion y \w:set.ordsucc (SNoLev w)) (famunion z \w:set.ordsucc (SNoLev w)))) (ordsucc x) -> SNoLev (SNoCut y z) iIn ordsucc x