reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;
reserve f,g for Function,
  L for Sequence,
  F for Cardinal-Function;
reserve U1,U2,U for Universe;
reserve u,v for Element of U;

theorem
  A c= B iff UNIVERSE A c= UNIVERSE B
proof
  thus A c= B implies UNIVERSE A c= UNIVERSE B
  proof
    assume
A1: A c= B;
    assume not UNIVERSE A c= UNIVERSE B;
    then UNIVERSE B in UNIVERSE A by Th53;
    then B in A by Th70;
    hence contradiction by A1,ORDINAL1:5;
  end;
  assume
A2: UNIVERSE A c= UNIVERSE B;
  assume not A c= B;
  then B in A by ORDINAL1:16;
  then UNIVERSE B in UNIVERSE A by Th70;
  hence contradiction by A2,ORDINAL1:5;
end;
