const TransSet : set prop const In : set set prop term iIn = In infix iIn 2000 2000 term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const PNo_rel_strict_upperbd : (set (set prop) prop) set (set prop) prop const PNo_rel_strict_lowerbd : (set (set prop) prop) set (set prop) prop term PNo_rel_strict_imv = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_rel_strict_upperbd P x p & PNo_rel_strict_lowerbd Q x p const PNoEq_ : set (set prop) (set prop) prop term PNo_rel_strict_uniq_imv = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_rel_strict_imv P Q x p & !q:set prop.PNo_rel_strict_imv P Q x q -> PNoEq_ x p q const ordsucc : set set axiom ordsuccI2: !x:set.x iIn ordsucc x const PNo_lenbdd : set (set (set prop) prop) prop const PNo_rel_strict_split_imv : (set (set prop) prop) (set (set prop) prop) set (set prop) prop axiom PNo_lenbdd_strict_imv_split: !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.ordinal x -> PNo_lenbdd x P -> PNo_lenbdd x Q -> !p:set prop.PNo_rel_strict_imv P Q x p -> PNo_rel_strict_split_imv P Q x p axiom ordsuccI1: !x:set.Subq x (ordsucc x) const PNoLt_pwise : (set (set prop) prop) (set (set prop) prop) prop axiom PNo_rel_imv_ex: !P:set (set prop) prop.!Q:set (set prop) prop.PNoLt_pwise P Q -> !x:set.ordinal x -> (?p:set prop.PNo_rel_strict_uniq_imv P Q x p) | ?y:set.y iIn x & ?p:set prop.PNo_rel_strict_split_imv P Q y p var P:set (set prop) prop var Q:set (set prop) prop var x:set hyp PNoLt_pwise P Q hyp ordinal x hyp PNo_lenbdd x P hyp PNo_lenbdd x Q claim ordinal (ordsucc x) -> ?y:set.y iIn ordsucc x & ?p:set prop.PNo_rel_strict_split_imv P Q y p