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 term nIn = \x:set.\y:set.~ x iIn y const SNo : set prop 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 ordinal : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.ordinal x -> ordinal y -> SNo (x + y) -> ordinal (SNoLev (x + y)) -> ordinal (x + y) var x:set var y:set hyp ordinal x hyp ordinal y hyp SNo x hyp SNo y claim SNo (x + y) -> ordinal (x + y)