Question about a subset Why is this thing true
 $$\{\{\varnothing\}\}\subset\{\varnothing,\{\varnothing\}\}$$
I'm trying to underestand, but I need an explanation.
 A: Let $A$ be a set; then $\{ \{ A \} \} \subset \{ A, \{ A \} \}$, for $\{ A \}$ belongs to the latter class. If you understand this,  take $A := \varnothing$ and your problem is solved.
A: If $x \in \{\{\varnothing\}\}$ then $x = \{\varnothing\}$
Since $ x \in \{\varnothing,x\}$ then $ x \in \{\varnothing,\{\varnothing\}\}$
Hence $x \in \{\{\varnothing\}\}$ implies $x \in \{\varnothing,\{\varnothing\}\}$
So $\{\{\varnothing\}\}\subseteq\{\varnothing,\{\varnothing\}\}$
A: {{$\varnothing$}} is a set that has one element in it.  Its element is {$\varnothing$}; a set that contains only the empty set as a member.
{$\varnothing$, {$\varnothing$}} is a set with two elements in it.  The first element is the empty set.  The second element is the same element above:  The set {$\varnothing$}; the set that contains only the empty set as a member.
Both sets contain the element {$\varnothing$}
Every element in the first set (it has only one) is also a member of the second set (it has two).  So the first set is a subset of the second.
A: Referring to the definition of the inclusion: a set $A$ is included in another set $B$ if and only if all elements of set $A$ belongs to set $B$.
Just apply it to $$A=\{\{\varnothing\}\}, \ B=\{\varnothing,\{\varnothing\}\}.$$ What are the elements of $A=\{\{\varnothing\}\}$? There is only one which is $\{\varnothing\}$. Is $\{\varnothing\}$ an element of $B$? The answer is yes.
Conclusion: $$A \subset B.$$
