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 nat_p : set prop const SNo : set prop axiom nat_p_SNo: !x:set.nat_p x -> SNo x const exp_SNo_nat : set set set const ordsucc : set set const Empty : set const Sep : set (set prop) set const SNoS_ : set set const omega : set const SNoLev : set set const binintersect : set set set const SNoElts_ : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!f:set set.!f2:set set.!y:set.nat_p x -> nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> (!z:set.z iIn exp_SNo_nat (ordsucc (ordsucc Empty)) x -> ?w:set.w iIn Sep (SNoS_ omega) (\u:set.SNoLev u = x) & f w = z) -> (!z:set.x iIn z -> f2 z = f (binintersect z (SNoElts_ x))) -> (!z:set.nIn x z -> f2 z = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect z (SNoElts_ x))) -> y iIn exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x -> nat_p y -> SNo y -> ?z:set.z iIn Sep (SNoS_ omega) (\w:set.SNoLev w = ordsucc x) & f2 z = y var x:set var f:set set var f2:set set var y:set hyp nat_p x hyp nat_p (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 !z:set.z iIn exp_SNo_nat (ordsucc (ordsucc Empty)) x -> ?w:set.w iIn Sep (SNoS_ omega) (\u:set.SNoLev u = x) & f w = z hyp !z:set.x iIn z -> f2 z = f (binintersect z (SNoElts_ x)) hyp !z:set.nIn x z -> f2 z = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect z (SNoElts_ x)) hyp y iIn exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x claim nat_p y -> ?z:set.z iIn Sep (SNoS_ omega) (\w:set.SNoLev w = ordsucc x) & f2 z = y