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 TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y axiom FalseE: ~ False const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe_or: !x:set.!y:set.SNo x -> SNo y -> x < y | y <= x const SNoR : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 var x:set var y:set var z:set var w:set hyp SNo y hyp SNo z hyp ~ ((?u:set.u iIn SNoR x & (u + y) <= z) | ?u:set.u iIn SNoR y & (x + u) <= z) hyp w iIn SNoR x hyp SNo w claim SNo (w + y) -> z < w + y