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 term nIn = \x:set.\y:set.~ x iIn y const SNo : set prop axiom ordinal_SNo: !x:set.ordinal x -> SNo x const ordsucc : set set axiom ordsuccI2: !x:set.x iIn ordsucc x const SNoLev : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom ordinal_SNoLev_max_2: !x:set.ordinal x -> !y:set.SNo y -> SNoLev y iIn ordsucc x -> y <= x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_Le2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y <= z -> (x + y) <= x + z const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom ordinal_In_SNoLt: !x:set.ordinal x -> !y:set.y iIn x -> y < x axiom add_SNo_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z axiom ordinal_SNoLt_In: !x:set.!y:set.ordinal x -> ordinal y -> x < y -> x iIn y axiom ordinal_ordsucc_In: !x:set.ordinal x -> !y:set.y iIn x -> ordsucc y iIn ordsucc x axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z var x:set var y:set var z:set hyp ordinal y hyp SNo x hyp SNo y hyp ordinal (x + y) hyp SNo (ordsucc x) hyp ordinal (ordsucc (x + y)) hyp SNo (ordsucc (x + y)) hyp SNoLev z iIn y hyp ordinal (SNoLev z) hyp SNo z hyp ordsucc x + SNoLev z = ordsucc (x + SNoLev z) hyp SNo (ordsucc x + z) hyp ordinal (x + SNoLev z) hyp ordinal (ordsucc x + SNoLev z) claim SNo (ordsucc x + SNoLev z) -> (ordsucc x + z) < ordsucc (x + y)