const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x 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 term PNo_least_rep2 = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_least_rep P Q x p & !y:set.nIn y x -> ~ p y term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) 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 axiom least_ordinal_ex: !p:set prop.(?x:set.ordinal x & p x) -> ?x:set.ordinal x & p x & !y:set.y iIn x -> ~ p y lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.PNoLt_pwise P Q -> ordinal x -> PNo_lenbdd x P -> PNo_lenbdd x Q -> (?y:set.ordinal y & (?p:set prop.PNo_strict_imv P Q y p) & !z:set.z iIn y -> ~ ?p:set prop.PNo_strict_imv P Q z p) -> ?y:set.(?p:set prop.PNo_least_rep2 P Q y p) & !p:set prop.!q:set prop.PNo_least_rep2 P Q y p -> PNo_least_rep2 P Q y q -> p = q lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!y:set.!p:set prop.ordinal x -> y iIn ordsucc x -> PNo_strict_imv P Q y p -> ordinal (ordsucc x) -> ?z:set.ordinal z & ?q:set prop.PNo_strict_imv P Q z q 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 -> ?y:set.(?p:set prop.PNo_least_rep2 P Q y p) & !p:set prop.!q:set prop.PNo_least_rep2 P Q y p -> PNo_least_rep2 P Q y q -> p = q