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 const ordinal : set prop const PNo_strict_imv : (set (set prop) prop) (set (set prop) prop) set (set prop) prop term PNo_least_rep = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.ordinal x & PNo_strict_imv P Q x p & !y:set.y iIn x -> !q:set prop.~ PNo_strict_imv P Q y q const ordsucc : set set axiom ordinal_ordsucc: !x:set.ordinal x -> ordinal (ordsucc x) const PNoLt_pwise : (set (set prop) prop) (set (set prop) prop) prop const PNo_lenbdd : set (set (set prop) prop) prop axiom PNo_lenbdd_strict_imv_ex: !P:set (set prop) prop.!Q:set (set prop) prop.PNoLt_pwise P Q -> !x:set.ordinal x -> PNo_lenbdd x P -> PNo_lenbdd x Q -> ?y:set.y iIn ordsucc x & ?p:set prop.PNo_strict_imv P Q y p const PNo_bd : (set (set prop) prop) (set (set prop) prop) set const PNo_pred : (set (set prop) prop) (set (set prop) prop) set prop axiom PNo_bd_pred: !P:set (set prop) prop.!Q:set (set prop) prop.PNoLt_pwise P Q -> !x:set.ordinal x -> PNo_lenbdd x P -> PNo_lenbdd x Q -> PNo_least_rep P Q (PNo_bd P Q) (PNo_pred P Q) lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!y:set.!p:set prop.ordinal x -> ordinal (PNo_bd P Q) -> (!z:set.z iIn PNo_bd P Q -> !q:set prop.~ PNo_strict_imv P Q z q) -> y iIn ordsucc x -> PNo_strict_imv P Q y p -> ordinal (ordsucc x) -> PNo_bd P Q iIn ordsucc x claim !P:set (set prop) prop.!Q:set (set prop) prop.PNoLt_pwise P Q -> !x:set.ordinal x -> PNo_lenbdd x P -> PNo_lenbdd x Q -> PNo_bd P Q iIn ordsucc x