:: Tarski {G}rothendieck Set Theory -- Tarski's Axiom A
:: by Andrzej Trybulec
::
:: Received January 1, 1989
:: Copyright (c) 1990-2026 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;
notations TARSKI;
constructors TARSKI;
begin
reserve x,y,z,u for object;
reserve N,M,X,Y,Z for set;
:: 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);