From Discrete and Combinatorial Mathematics by Ralph P. Grimaldi, 5th ed., on Set Theory:
Let U = {1,2,3,4,5,6,x,y, {1,2}, {1,2,3}, {1,2,3,4}}. Then |U| = 11.
If A = {1,2,3,4}, then |A| = 4 and we have
1) A is a subset of U
2) A is a proper subset of U
3) A is an element of U
4) {A} is a subset of U
5) {A} is a proper subset of U
6) BUT {A} is not an element of U
Why is {A} is not an element of U? Isn't {A}={1,2,3,4} considered an element in U?
Best Answer
Note that $\{A\}=\{\{1,2,3,4\}\}$
Now we have $\{1,2,3,4\}\in U$ so all the elements of $\{A\}$ are elements of $U$, which is essentially the definition of $\{A\}\subset U$
But we don't have $\{\{1,2,3,4\}\}\in U$ so we don't have $\{A\}\in U$.