For any set $A$, let $T(A)$ be the set consisting of all sets $S \subseteq \mathcal{P}(A)$ that satisfy the following conditions:
- (i) $\emptyset \in S$
- (ii) $A \in S$
- (iii) $\forall X, Y \in S, (X \cup Y) \in S$
- (iv) $\forall X, Y \in S, (X \cap Y) \in S$
What are the number of elements of the set $\{S \in T(\{1, 2, 3\}) : |S| \geqslant 5\}$?
I'm not sure how to go about this. Handling these extra conditions makes finding the size of tis set difficult for me.