@MISC{ZF_FUND1.MIZ,
  AUTHOR = {Kondracki, Andrzej},
  TITLE = {Mostowski's Fundamental Operations --- Part {I}},
  DAY = {17},
  MONTH = {December},
  YEAR = {1990},
  ADDRESS1 = {Warsaw University},
  SUMMARY = {In the chapter II.4 of his book \cite {MOST:1} A.~Mostowski introduces
what he calls fundamental operations:\\
 \indent $A_{1}(a,b)=\lbrace\lbrace\langle0,x\rangle,\langle1,y\rangle\rbrace:
x\in y \wedge x\in a \wedge y\in a \rbrace$,\\
 \indent $A_{2}(a,b)=\lbrace a,b\rbrace$,\\
 \indent $A_{3}(a,b)=\bigcup a$,\\
 \indent $A_{4}(a,b)=\lbrace\lbrace\langle x,y\rangle\rbrace:
x\in a \wedge y\in b \rbrace$,\\
 \indent $A_{5}(a,b)=\lbrace x\cup y : x\in a \wedge y\in b \rbrace$,\\
 \indent $A_{6}(a,b)=\lbrace x\setminus y : x\in a \wedge y\in b \rbrace$,\\
 \indent $A_{7}(a,b)=\lbrace x\circ y : x\in a \wedge y\in b \rbrace$.\\
He proves that if a non-void class is closed under these operations then
it is predicatively closed. Then he formulates sufficient criteria for a class
to be a model of ZF set theory (theorem 4.12). \par
   The article includes the translation of this part of Mostowski's book.
The fundamental operations are defined (to be precise not these operations, but
the notions of closure of a class with respect to them). Some properties of
classes closed under these operations are proved. At last it is proved that
if a non-void class $X$ is closed with respect to the operations $A_{1}-A_{7}$
then $D_{H}(a)\in X$ for every $a$ in $X$ and every $H$ being formula of
ZF language ($D_{H}(a)$ consists of all finite sequences with terms belonging
to $a$ which satisfy $H$ in $a$).}}

