const ordinal : set prop const SNo_ : set set prop term SNo = \x:set.?y:set.ordinal y & SNo_ y x const In : set set prop term iIn = In infix iIn 2000 2000 const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w const Sep : set (set prop) set const SNoS_ : set set const SNoLev : set set term SNoR = \x:set.Sep (SNoS_ (SNoLev x)) (SNoLt x) term SNoL = \x:set.Sep (SNoS_ (SNoLev x)) \y:set.y < x axiom SepE2: !x:set.!p:set prop.!y:set.y iIn Sep x p -> p y axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z lemma !x:set.SNo x -> (!y:set.y iIn SNoL x -> SNo y) -> (!y:set.y iIn SNoR x -> SNo y) -> (!y:set.y iIn SNoL x -> !z:set.z iIn SNoR x -> y < z) -> (!y:set.y iIn SNoL x -> SNo y) & (!y:set.y iIn SNoR x -> SNo y) & !y:set.y iIn SNoL x -> !z:set.z iIn SNoR x -> y < z var x:set hyp SNo x hyp ordinal (SNoLev x) hyp !y:set.y iIn SNoL x -> SNo y claim (!y:set.y iIn SNoR x -> SNo y) -> (!y:set.y iIn SNoL x -> SNo y) & (!y:set.y iIn SNoR x -> SNo y) & !y:set.y iIn SNoL x -> !z:set.z iIn SNoR x -> y < z