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 ordsucc : set set axiom ordsuccI2: !x:set.x iIn ordsucc x const SNo : set prop const exp_SNo_nat : set 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 const nat_p : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 lemma !x:set.!f:set set.!f2:set set.!y:set.!z:set.SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> (!w:set.w iIn Sep (SNoS_ omega) (\u:set.SNoLev u = x) -> !u:set.u iIn Sep (SNoS_ omega) (\v:set.SNoLev v = x) -> f w = f u -> w = u) -> (!w:set.x iIn w -> f2 w = f (binintersect w (SNoElts_ x))) -> (!w:set.nIn x w -> f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x))) -> SNo y -> SNoLev y = ordsucc x -> binintersect y (SNoElts_ x) iIn Sep (SNoS_ omega) (\w:set.SNoLev w = x) -> nat_p (f (binintersect y (SNoElts_ x))) -> SNo (f (binintersect y (SNoElts_ x))) -> f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x -> SNo z -> SNoLev z = ordsucc x -> binintersect z (SNoElts_ x) iIn Sep (SNoS_ omega) (\w:set.SNoLev w = x) -> nat_p (f (binintersect z (SNoElts_ x))) -> SNo (f (binintersect z (SNoElts_ x))) -> f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x -> x iIn SNoLev y -> x iIn SNoLev z -> f2 y = f2 z -> y = z var x:set var f:set set var f2:set set var y:set var z:set hyp SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x) hyp !w:set.w iIn Sep (SNoS_ omega) (\u:set.SNoLev u = x) -> !u:set.u iIn Sep (SNoS_ omega) (\v:set.SNoLev v = x) -> f w = f u -> w = u hyp !w:set.x iIn w -> f2 w = f (binintersect w (SNoElts_ x)) hyp !w:set.nIn x w -> f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)) hyp SNo y hyp SNoLev y = ordsucc x hyp binintersect y (SNoElts_ x) iIn Sep (SNoS_ omega) \w:set.SNoLev w = x hyp nat_p (f (binintersect y (SNoElts_ x))) hyp SNo (f (binintersect y (SNoElts_ x))) hyp f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x hyp SNo z hyp SNoLev z = ordsucc x hyp binintersect z (SNoElts_ x) iIn Sep (SNoS_ omega) \w:set.SNoLev w = x hyp nat_p (f (binintersect z (SNoElts_ x))) hyp SNo (f (binintersect z (SNoElts_ x))) hyp f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x claim x iIn SNoLev y -> f2 y = f2 z -> y = z