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;
