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) axiom iffI: !P:prop.!Q:prop.(P -> Q) -> (Q -> P) -> (P <-> Q) axiom PNoEq_strict_imv: !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.ordinal x -> !p:set prop.!q:set prop.PNoEq_ x p q -> PNo_strict_imv P Q x p -> PNo_strict_imv P Q x q const PNoLt_pwise : (set (set prop) prop) (set (set prop) prop) prop axiom PNo_strict_imv_pred_eq: !P:set (set prop) prop.!Q:set (set prop) prop.PNoLt_pwise P Q -> !x:set.ordinal x -> !p:set prop.!q:set prop.PNo_least_rep P Q x p -> PNo_strict_imv P Q x q -> !y:set.y iIn x -> (p y <-> q y) axiom FalseE: ~ False axiom xm: !P:prop.P | ~ P axiom pred_ext: !p:set prop.!q:set prop.(!x:set.p x <-> q x) -> p = q const PNo_lenbdd : set (set (set prop) prop) prop 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 (?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