:: Tarski {G}rothendieck Set Theory
::  by Andrzej Trybulec
::
:: Received January 1, 1989
:: Copyright (c) 1990-2012 Association of Mizar Users
::           (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.

environ

 vocabularies TARSKI;

begin

 reserve x,y,z,u,N,M,X,Y,Z for set;

theorem :: Extensionality
 (for x holds x in X iff x in Y) implies X = Y;

 definition let y; func { y } -> set means
  x in it iff x = y;
  correctness;
  let z; func { y, z } -> set means
   x in it iff x = y or x = z;
   correctness;
   commutativity;
 end;

 definition let X,Y;
  pred X c= Y means
   x in X implies x in Y;
  reflexivity;
 end;

 definition let X;
   func union X -> set means
    x in it iff ex Y st x in Y & Y in X;
   correctness;
 end;

theorem :: Regularity
 x in X implies ex Y st Y in X & not ex x st x in X & x in Y;

 scheme Fraenkel { A()-> set, P[set, set] }:
  ex X st for x holds x in X iff ex y st y in A() & P[y,x]
  provided for x,y,z st P[x,y] & P[x,z] holds y = z
  proof
    thus thesis;
  end;

  definition let x,y;
  func [x,y] equals
   { { x,y }, { x } };
   correctness;
 end;

 definition let X,Y;
  pred X,Y are_equipotent means
    ex Z st
       (for x st x in X ex y st y in Y & [x,y] in Z) &
       (for y st y in Y ex x st x in X & [x,y] in Z) &
        for x,y,z,u st [x,y] in Z & [z,u] in Z holds x = z iff y = u;
 end;

:: Alfred Tarski
:: Ueber unerreichbare Kardinalzahlen,
:: Fundamenta Mathematicae, vol. 30 (1938), pp. 68--69

:: Axiom A. (Axiom der unerreichbaren Mengen). Zu jeder Menge N gibt es
:: eine Menge M mit folgenden Eigenschaften:
:: A1. N in M;
:: A2. ist X in M und Y c= X, so ist Y in M;
:: A3. ist X in M und ist Z die Menge, die alle Mengen Y c= X und keine
::     andere Dinge als Element enthaelt, so,ist z in M;
:: A4. ist X c= M und sind dabei die Menge X und M nicht gleichmaechtig,
::     so ist X in M.

:: Alfred Tarski
:: On Well-ordered Subsets of any Set,
:: Fundamenta Mathematicae, vol. 32 (1939), pp. 176--183

:: A. For every set N there exists a system M of sets which satisfies
::    the following conditions:
:: (i)    N in M
:: (ii)   if X in M and Y c= X, then Y in M
:: (iii)  if X in M and Z is the system of all subsets of X, then Z in M
:: (iv)   if X c= M and X and M do not have the same potency, then X in M.

theorem
  ex M st N in M &
     (for X,Y holds X in M & Y c= X implies Y in M) &
     (for X st X in M ex Z st Z in M & for Y st Y c= X holds Y in Z) &
     (for X holds X c= M implies X,M are_equipotent or X in M);