Statement
The sets of all equivalence relations and of all partitions does not exist.
Proof. First of all we observe that the idetity relations $\text{Id}$ is an equivalence relations for any set $A$ and additionaly the corresponding quotient set $A/\text{Id}$ is the same set $A$; then we remember that any equivalence relation is completely determined by the quotient set. So if there exist the set of all equivalence relations then there exist the set of all quotient set that, for what we have observed above, contains the set of all sets and this is clearly impossible. Now we know that any partition determine an equivalence relation and vice versa. So if there exist the set of all partition then there exist the set of all equivalence relation too and, for what we have proved above, this is impossible.
So I ask if the statement if true and so if my proof is correct and if it is possible to improve it. So could someone help me, please?