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 nIn = \x:set.\y:set.~ x iIn y const binunion : set set set const Repl : set (set set) set const SetAdjoin : set set set const Sing : set set const ordsucc : set set const Empty : set term SNoElts_ = \x:set.binunion x (Repl x \y:set.SetAdjoin y (Sing (ordsucc Empty))) term SNo_ = \x:set.\y:set.Subq y (SNoElts_ x) & !z:set.z iIn x -> ~(SetAdjoin z (Sing (ordsucc Empty)) iIn y <-> z iIn y) axiom exactly1of2_E: !P:prop.!Q:prop.~(P <-> Q) -> !R:prop.(P -> ~ Q -> R) -> (~ P -> Q -> R) -> R axiom FalseE: ~ False const SNoLev : set set var x:set var y:set var z:set var w:set hyp Subq (SNoLev x) (SNoLev y) hyp !u:set.u iIn SNoLev x -> (u iIn x <-> u iIn y) hyp !u:set.u iIn SNoLev x -> ~(SetAdjoin u (Sing (ordsucc Empty)) iIn x <-> u iIn x) hyp !u:set.u iIn SNoLev y -> ~(SetAdjoin u (Sing (ordsucc Empty)) iIn y <-> u iIn y) hyp z iIn x hyp w iIn SNoLev x hyp z = SetAdjoin w (Sing (ordsucc Empty)) claim w iIn SNoLev y -> z iIn y