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 TransSet = \x:set.!y:set.y iIn x -> Subq y x const SNo : set prop const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 term SNo_max_of = \x:set.\y:set.y iIn x & SNo y & !z:set.z iIn x -> SNo z -> z <= y const ordinal : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom ordinal_TransSet: !x:set.ordinal x -> TransSet x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const SNoR : set set axiom SNoR_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> x < y -> y iIn SNoR x axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoS_ : set set var x:set var y:set var z:set var w:set hyp SNo x hyp SNo z hyp !u:set.u iIn SNoS_ (SNoLev z) -> x < u -> (y + u) < x + x -> ?v:set.v iIn SNoR u & y + v = x + x hyp x < z hyp w iIn SNoR z hyp (y + w) < x + x hyp SNo w hyp SNoLev w iIn SNoLev z hyp z < w claim (?u:set.u iIn SNoR w & y + u = x + x) -> ?u:set.u iIn SNoR z & y + u = x + x