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 ordsucc : set set axiom ordsuccI2: !x:set.x iIn ordsucc x axiom In_no2cycle: !x:set.!y:set.x iIn y -> ~ y iIn x const famunion : set (set set) set axiom famunionE_impred: !x:set.!f:set set.!y:set.y iIn famunion x f -> !P:prop.(!z:set.z iIn x -> y iIn f z -> P) -> P const binunion : set set set axiom binunionE: !x:set.!y:set.!z:set.z iIn binunion x y -> z iIn x | z iIn y axiom FalseE: ~ False const ordinal : set prop axiom ordinal_In_Or_Subq: !x:set.!y:set.ordinal x -> ordinal y -> x iIn y | Subq y x axiom ordinal_ordsucc_In_Subq: !x:set.ordinal x -> !y:set.y iIn x -> Subq (ordsucc y) x const SNoCutP : set set prop const SNoLev : set set const SNoCut : set set set lemma !x:set.!y:set.!z:set.(!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) -> 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 var x:set var y:set var z:set hyp ordinal x 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))) claim ordinal (ordsucc x) -> SNoLev (SNoCut y z) iIn ordsucc x