Is $\{\varnothing\}\in\{\varnothing,\{\varnothing\}\}$ true?

Is {∅} ∈ {∅, {∅}} true?

YES.

Can we use the same explanation as in {∅} ⊆ {∅, {∅}}?

NO.

• You prove that $$\{\emptyset\}\in \{\emptyset, \{\emptyset\}\}$$ by proving that $\{\emptyset\}$ is an element of $\{\emptyset, \{\emptyset\}\}$. This is fairly obvious, and is the same as proving that $0$ is an element of $\{0,1\}$.
• You prove that $$\{\emptyset\}\subseteq \{\emptyset, \{\emptyset\}\}$$ by showing that every element of $\{\emptyset\}$ is also an element of $\{\emptyset, \{\emptyset\}\}$. Since $\{\emptyset\}$ has only one element, $\emptyset$, this means you need to prove that $\emptyset$ is an element of $\{\emptyset,\{\emptyset\}\}$.

Can we say the same thing for {∅} ∈ {∅, {{∅}}}, and {{∅}} ∈ {∅, {∅}}?

NO.

Remember:

$A$ is a subset of $B$ (denoted as $A\subseteq B$) if and only if for every $a\in A$, it is true that $a\in B$.

This means that:

• $\{\emptyset\}\in \{\emptyset, \{\emptyset\}\}$ is true the same way $1\in \{0,1\}$ is true.
• $\{\emptyset\} \subset \{\emptyset,\{\emptyset\}\}$ is true the same way $\{a\}\subset \{a,b\}$ is true.
• $\{\{\emptyset\}\} \subset \{\emptyset,\{\emptyset\}\}$ is true, because every element of $\{\{\emptyset\}\}$ (there is only one) is an element of $\{\emptyset,\{\emptyset\}\}$.
• $\{\{\emptyset\}\}\in \{\emptyset, \{\emptyset\}\}$ is false, because neither of the two elements in $\{\emptyset,\{\emptyset\}\}$ is equal to $\{\{\emptyset\}\}$.

The easiest way to resolve some of your queries is by introducing convenient notation so as not to get lost in the curly braces, such as $0=\emptyset$, $1=\{\emptyset\}$, $2=\{\emptyset,\{\emptyset\}\}=\{0,1\}$, etc. For example, the answer to your last question is negative because it is not true that $\{1\}\in\{0,1\}$, only that $\{1\}\subset\{0,1\}$.