
theorem Th26:
for G being finite SimpleGraph holds order G <= card G
proof
  let G be finite SimpleGraph;
  set uG = union G;
A1: card singletons uG = card uG by BSPACE:41;
    singletons uG c= G proof
     let x be object;
     assume x in singletons uG;
     then consider f being Subset of uG such that
A2: x = f and
A3: f is 1-element;
     consider a being set such that
A4: a in uG and
A5: f = {a} by A3,Th9;
     consider y being set such that
A6: a in y and
A7: y in G by A4,TARSKI:def 4;
     {a} c= y by A6,ZFMISC_1:31;
     hence x in G by A7,A5,A2,CLASSES1:def 1;
    end;
  hence order G <= card G by A1,NAT_1:43;
end;
