reserve A,B,a,b,c,d,e,f,g,h for set;

theorem Th5:
  for X be 1-element set, R be Relation of X holds R is_symmetric_in X
proof
  let X be 1-element set;
  let R be Relation of X;
  consider x be object such that
A1: X = {x} by ZFMISC_1:131;
  let a,b be object;
  assume that
A2: a in X and
A3: b in X & [a,b] in R;
  a = x by A1,A2,TARSKI:def 1;
  hence thesis by A1,A3,TARSKI:def 1;
end;
