Perhaps empty relation is not well defined.  Define a relation ~ on the integers such that for all integers a and b, a~b is false.  2 and 3 are vacuously satisfied but 1~1 is false, so this is not an equivalence relation.  The error in your proof is thus the assumption that there exists a, c such that a=c.  If there is (well, if for all a there exists c such that a~c) then we have an equivalence relation.