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 const PNoLe : set (set prop) set (set prop) prop term PNo_upc = \P:set (set prop) prop.\x:set.\p:set prop.?y:set.ordinal y & ?q:set prop.P y q & PNoLe y q x p const PNoLt : set (set prop) set (set prop) prop term PNo_rel_strict_lowerbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.y iIn x -> !q:set prop.PNo_upc P y q -> PNoLt x p y q term PNo_downc = \P:set (set prop) prop.\x:set.\p:set prop.?y:set.ordinal y & ?q:set prop.P y q & PNoLe x p y q term PNo_rel_strict_upperbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.y iIn x -> !q:set prop.PNo_downc P y q -> PNoLt y q x p 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 term PNo_lenbdd = \x:set.\P:set (set prop) prop.!y:set.!p:set prop.P y p -> y iIn x const PNoEq_ : set (set prop) (set prop) prop term PNoLt_ = \x:set.\p:set prop.\q:set prop.?y:set.y iIn x & (PNoEq_ y p q & ~ p y & q y) term nIn = \x:set.\y:set.~ x iIn y const ordsucc : set set axiom ordsuccI2: !x:set.x iIn ordsucc x axiom PNoEq_sym_: !x:set.!p:set prop.!q:set prop.PNoEq_ x p q -> PNoEq_ x q 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 PNo_upc_ref: !P:set (set prop) prop.!x:set.ordinal x -> !p:set prop.P x p -> PNo_upc P x p axiom PNoLt_tra: !x:set.!y:set.!z:set.ordinal x -> ordinal y -> ordinal z -> !p:set prop.!q:set prop.!p2:set prop.PNoLt x p y q -> PNoLt y q z p2 -> PNoLt x p z p2 lemma !P:set (set prop) prop.!x:set.!p:set prop.!y:set.!q:set prop.!z:set.!p2:set prop.ordinal x -> PNo_lenbdd x P -> PNo_rel_strict_lowerbd P x p -> ordinal y -> ordinal z -> P z p2 -> PNoLe z p2 y q -> PNo_upc P z p2 -> PNoLt x p y q var P:set (set prop) prop var x:set var p:set prop var y:set var q:set prop hyp ordinal x hyp PNo_lenbdd x P hyp PNo_rel_strict_lowerbd P x p hyp ordinal (ordsucc x) hyp PNoEq_ x p \z:set.p z & z != x hyp y iIn ordsucc x hyp PNo_upc P y q claim ordinal y -> PNoLt (ordsucc x) (\z:set.p z & z != x) y q