const bij : set set (set set) prop term equip = \x:set.\y:set.?f:set set.bij x y f const In : set set prop term iIn = In infix iIn 2000 2000 term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) term nIn = \x:set.\y:set.~ x iIn y term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const ordsucc : set set axiom ordsuccI2: !x:set.x iIn ordsucc x const SNoS_ : set set const SNoLev : set set const SNo : set prop const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P const Sep : set (set prop) set axiom SepE: !x:set.!p:set prop.!y:set.y iIn Sep x p -> y iIn x & p y axiom bijE: !x:set.!y:set.!f:set set.bij x y f -> !P:prop.((!z:set.z iIn x -> f z iIn y) -> (!z:set.z iIn x -> !w:set.w iIn x -> f z = f w -> z = w) -> (!z:set.z iIn y -> ?w:set.w iIn x & f w = z) -> P) -> P const nat_p : set prop const exp_SNo_nat : set set set const Empty : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const binintersect : set set set const SNoElts_ : set set lemma !x:set.!f:set set.nat_p x -> nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> (!y:set.SNo y -> y < exp_SNo_nat (ordsucc (ordsucc Empty)) x -> (exp_SNo_nat (ordsucc (ordsucc Empty)) x + y) < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> exp_SNo_nat (ordsucc (ordsucc Empty)) x < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.SNoLev z = x) -> f y iIn exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.SNoLev z = x) -> !z:set.z iIn Sep (SNoS_ omega) (\w:set.SNoLev w = x) -> f y = f z -> y = z) -> (!y:set.y iIn exp_SNo_nat (ordsucc (ordsucc Empty)) x -> ?z:set.z iIn Sep (SNoS_ omega) (\w:set.SNoLev w = x) & f z = y) -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.SNoLev z = ordsucc x) -> !P:prop.(SNo y -> SNoLev y = ordsucc x -> binintersect y (SNoElts_ x) iIn Sep (SNoS_ omega) (\z:set.SNoLev z = x) -> SNo (binintersect y (SNoElts_ x)) -> SNoLev (binintersect y (SNoElts_ x)) = x -> P) -> P) -> equip (Sep (SNoS_ omega) \y:set.SNoLev y = ordsucc x) (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) lemma !x:set.!y:set.!P:prop.nat_p x -> SNoLev y = ordsucc x -> SNo y -> (SNo y -> SNoLev y = ordsucc x -> binintersect y (SNoElts_ x) iIn Sep (SNoS_ omega) (\z:set.SNoLev z = x) -> SNo (binintersect y (SNoElts_ x)) -> SNoLev (binintersect y (SNoElts_ x)) = x -> P) -> x iIn SNoLev y -> P var x:set hyp nat_p x hyp equip (Sep (SNoS_ omega) \y:set.SNoLev y = x) (exp_SNo_nat (ordsucc (ordsucc Empty)) x) hyp nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x) hyp ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x) hyp SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x) hyp nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) hyp ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) hyp SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) hyp !y:set.SNo y -> y < exp_SNo_nat (ordsucc (ordsucc Empty)) x -> (exp_SNo_nat (ordsucc (ordsucc Empty)) x + y) < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x claim exp_SNo_nat (ordsucc (ordsucc Empty)) x < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x -> equip (Sep (SNoS_ omega) \y:set.SNoLev y = ordsucc x) (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)