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 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 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 const SNoL : set set axiom SNoL_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> y < x -> y iIn SNoL x axiom SNoLtLe_or: !x:set.!y:set.SNo x -> SNo y -> x < y | y <= x axiom add_SNo_Lt3b: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> x <= z -> y < w -> (x + y) < z + w axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z axiom SNoLt_irref: !x:set.~ x < x const SNoR : set set 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 SNoS_ : set set lemma !x:set.!y:set.!z:set.SNo x -> SNo y -> (!w:set.w iIn SNoL x -> SNo w -> w <= y) -> SNoLev y iIn SNoLev x -> SNo z -> (!w:set.w iIn SNoS_ (SNoLev z) -> x < w -> (y + w) < x + x -> ?u:set.u iIn SNoR w & y + u = x + x) -> x < z -> (y + z) < x + x -> SNo (x + x) -> SNo (y + z) -> (!w:set.w iIn SNoR y -> ~ (w + z) <= x + x) -> ?w:set.w iIn SNoR z & y + w = x + x lemma !x:set.!y:set.!z:set.SNo x -> SNo y -> (!w:set.w iIn SNoL x -> SNo w -> w <= y) -> SNoLev y iIn SNoLev x -> SNo z -> SNoLev z iIn SNoLev y -> y < z -> z < x -> z iIn SNoL x -> x <= z var x:set var y:set var z:set hyp SNo x hyp SNo y hyp !w:set.w iIn SNoL x -> SNo w -> w <= y hyp SNoLev y iIn SNoLev x hyp SNo z hyp !w:set.w iIn SNoS_ (SNoLev z) -> x < w -> (y + w) < x + x -> ?u:set.u iIn SNoR w & y + u = x + x hyp x < z hyp (y + z) < x + x hyp SNo (x + x) claim SNo (y + z) -> ?w:set.w iIn SNoR z & y + w = x + x