theorem
  A c= B iff aleph A c= aleph B
proof
A1: aleph A c< aleph B iff aleph A <> aleph B & aleph A c= aleph B;
  A in B iff aleph A in aleph B by Th20;
  hence thesis by A1,Th21,ORDINAL1:11,XBOOLE_0:def 8;
end;
