@MISC{ZF_FUND2.MIZ,
  AUTHOR = {Bancerek, Grzegorz and Kondracki, Andrzej},
  TITLE = {Mostowski's Fundamental Operations --- Part {II}},
  DAY = {15},
  MONTH = {February},
  YEAR = {1991},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  ADDRESS2 = {Warsaw University},
  SUMMARY = {The article consists of two parts. The first part is translation of
chapter II.3 of \cite{MOST:1}.
A section of $D_{H}(a)$ determined by $f$ (symbolically
 $S_{H}(a,f)$) and a notion of predicative
closure of a class are defined. It is proved that if following assumptions
are satisfied:
  (o)  $A=\bigcup_{\xi}A_{\xi}$,
  (i)     $A_{\xi} \subset A_{\eta}$ for $\xi < \eta$,
  (ii)    $A_{\lambda}=\bigcup_{\xi<\lambda}A_{\lambda}$
                 ($\lambda$ is a limit number),
  (iii)   $A_{\xi}\in A$,
  (iv)    $A_{\xi}$ is transitive,
  (v)     $(x,y\in A) \rightarrow (x\cap y\in A)$,
  (vi)    $A$ is predicatively closed,
then the axiom of power sets and the axiom of substitution are valid in $A$.
The second part is continuation of \cite{ZF_FUND1.ABS}. It is proved that if
a non-void, transitive
class is closed with respect to the operations $A_{1}-A_{7}$ then
it is predicatively closed.
At last sufficient criteria for a class to be a model of ZF-theory
are formulated:
if $A_{\xi}$ satisfies o -- iv and $A$ is closed under
the operations $A_{1}-A_{7}$ then $A$ is a model of ZF.}}

