reserve H,S for ZF-formula,
  x for Variable,
  X,Y for set,
  i for Element of NAT,
  e,u for set;
reserve M,M1,M2 for non empty set,
  f for Function,
  v1 for Function of VAR,M1,
  v2 for Function of VAR,M2,
  F,F1,F2 for Subset of WFF,
  W for Universe,
  a,b,c for Ordinal of W,
  A,B,C for Ordinal,
  L for DOMAIN-Sequence of W,
  va for Function of VAR,L.a,
  phi,xi for Ordinal-Sequence of W;

theorem Th5:
  M |= ZF-axioms & M is epsilon-transitive implies M is being_a_model_of_ZF
proof
  the_axiom_of_power_sets in WFF by ZF_LANG:4;
  then
A1: the_axiom_of_power_sets in ZF-axioms by Def4;
  the_axiom_of_infinity in WFF by ZF_LANG:4;
  then
A2: the_axiom_of_infinity in ZF-axioms by Def4;
  the_axiom_of_unions in WFF by ZF_LANG:4;
  then
A3: the_axiom_of_unions in ZF-axioms by Def4;
  assume that
A4: for H st H in ZF-axioms holds M |= H and
A5: M is epsilon-transitive;
  the_axiom_of_pairs in WFF by ZF_LANG:4;
  then the_axiom_of_pairs in ZF-axioms by Def4;
  hence M is epsilon-transitive & M |= the_axiom_of_pairs & M |=
the_axiom_of_unions & M |= the_axiom_of_infinity & M |= the_axiom_of_power_sets
  by A4,A5,A3,A2,A1;
  let H;
  assume
A6: {x.0,x.1,x.2} misses Free H;
  the_axiom_of_substitution_for H in WFF by ZF_LANG:4;
  then the_axiom_of_substitution_for H in ZF-axioms by A6,Def4;
  hence thesis by A4;
end;
