src/ZF/Cardinal.thy

Addition of cardinals and order types, various tidying

(*  Title: 	ZF/Cardinal.thy
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

Cardinals in Zermelo-Fraenkel Set Theory 
*)

Cardinal = OrderType + Fixedpt + Nat + Sum + 
consts
  Least            :: "(i=>o) => i"    (binder "LEAST " 10)
  eqpoll, lepoll   :: "[i,i] => o"     (infixl 50)
  "cardinal"       :: "i=>i"           ("|_|")
  Card             :: "i=>o"

  swap       :: "[i,i,i]=>i"     (*not used; useful??*)

rules

  (*least ordinal operator*)
  Least_def  "Least(P) == THE i. Ord(i) & P(i) & (ALL j. j<i --> ~P(j))"

  eqpoll_def "A eqpoll B == EX f. f: bij(A,B)"

  lepoll_def "A lepoll B == EX f. f: inj(A,B)"

  cardinal_def "|A| == LEAST i. i eqpoll A"

  Card_def     "Card(i) == ( i = |i| )"

  swap_def   "swap(A,x,y) == (lam z:A. if(z=x, y, if(z=y, x, z)))"

end