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 term TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) const PNoLt : set (set prop) set (set prop) prop term PNo_strict_lowerbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.ordinal y -> !q:set prop.P y q -> PNoLt x p y q term PNo_strict_upperbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.ordinal y -> !q:set prop.P y q -> PNoLt y q x p term PNo_strict_imv = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_strict_upperbd P x p & PNo_strict_lowerbd Q x p 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 axiom PNoEq_sym_: !x:set.!p:set prop.!q:set prop.PNoEq_ x p q -> PNoEq_ x q p axiom PNoLtI2: !x:set.!y:set.!p:set prop.!q:set prop.x iIn y -> PNoEq_ x p q -> q x -> PNoLt x p y q axiom dneg: !P:prop.~ ~ P -> P axiom PNoLtI3: !x:set.!y:set.!p:set prop.!q:set prop.y iIn x -> PNoEq_ y p q -> ~ p y -> PNoLt x p y q axiom iffI: !P:prop.!Q:prop.(P -> Q) -> (Q -> P) -> (P <-> Q) lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!p:set prop.!q:set prop.!y:set.ordinal x -> (!z:set.z iIn x -> !p2:set prop.~ PNo_strict_imv P Q z p2) -> PNo_strict_lowerbd Q x p -> PNo_strict_upperbd P x q -> ordinal y -> y iIn x -> PNoEq_ y p q -> p y -> ~ q y -> ~ PNoLt y q x p lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!p:set prop.!q:set prop.!y:set.ordinal x -> (!z:set.z iIn x -> !p2:set prop.~ PNo_strict_imv P Q z p2) -> PNo_strict_upperbd P x p -> PNo_strict_lowerbd Q x q -> ordinal y -> y iIn x -> PNoEq_ y p q -> q y -> ~ p y -> ~ PNoLt x p y q var P:set (set prop) prop var Q:set (set prop) prop var x:set var p:set prop var q:set prop var y:set hyp ordinal x hyp TransSet x hyp !z:set.z iIn x -> !p2:set prop.~ PNo_strict_imv P Q z p2 hyp PNo_strict_upperbd P x p hyp PNo_strict_lowerbd Q x p hyp PNo_strict_upperbd P x q hyp PNo_strict_lowerbd Q x q hyp ordinal y hyp !z:set.z iIn y -> z iIn x -> (p z <-> q z) hyp y iIn x claim PNoEq_ y p q -> (p y <-> q y)