reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;
reserve f for Ordinal-Sequence;
reserve U,W for Universe;
reserve F,phi for normal Ordinal-Sequence of W;
reserve g for Ordinal-Sequence-valued Sequence;

theorem Th57:
  for U being Universe holds U is uncountable iff omega in U
  proof
    let U be Universe;
    thus U is uncountable implies omega in U
    proof assume
      card U c/= omega; then
      omega in card U by Th4; then
      omega in On U by CLASSES2:9;
      hence omega in U by ORDINAL1:def 9;
    end;
    assume omega in U; then
    card omega in card U by CLASSES2:1;
    hence card U c/= omega;
  end;
