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 term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe: !x:set.!y:set.x < y -> x <= y axiom SNoLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y <= z -> x <= z const SNoL : set set const SNoLev : set set axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const SNoS_ : set set var x:set var y:set var z:set var w:set hyp SNo x hyp SNo y hyp SNo (x + y) hyp SNo z hyp !u:set.u iIn SNoS_ (SNoLev z) -> SNoLev u iIn SNoLev (x + y) -> u < x + y -> (?v:set.v iIn SNoL x & u <= v + y) | ?v:set.v iIn SNoL y & u <= x + v hyp SNoLev z iIn SNoLev (x + y) hyp ~ ((?u:set.u iIn SNoL x & z <= u + y) | ?u:set.u iIn SNoL y & z <= x + u) hyp SNo w hyp SNoLev w iIn SNoLev z hyp z < w hyp w < x + y hyp w iIn SNoS_ (SNoLev z) claim ~ SNoLev w iIn SNoLev (x + y)