Conceptual Doubt with Sets regarding empty set

So the empty set $\emptyset$ is a subset of every set.

So $\emptyset \subset \{1,2,3\}$

But why isn't

$\{\emptyset\} \subset \{1,2,3,\{6,7\}\}$

Shouldn't it be valid because {..other objects..{6,7}} contain every object present in {${\emptyset}} which is nothing. and even if we enclose a empty set inside a set isn't it still going to create another empty set?$\{\emptyset\} = \{\{\}\}$is this equal to$\{\}$since there is nothing inside. Thanks for the help, •$\{\emptyset\} \subset \{1,2,3,\{6,7\}\}$would be true iff$\emptyset$is a member of$\{1,2,3,\{6,7\}\}$. – mrs Mar 23 '16 at 13:09 • @Clarinetist. No.$\{1\}$and$\{\{1\}\}$are two different sets. – almagest Mar 23 '16 at 15:11 • Suppose$x=\{\;\{\}\;\}$. This means$\forall y\;(y\in x\iff y=\{\}\;)$. So if$y$is the empty set then$ y\in x.$So$x$has a member ( namely,$\phi \in x$) .So$x$is not the empty set. – DanielWainfleet Mar 25 '16 at 20:50 2 Answers 1)$\{ \emptyset \} \not\subset \{1,2,3,\{6,7\}\}$, because$\emptyset \not \in \{1,2,3,\{6,7\}\}$. Indeed,$\emptyset \neq 1$,$\emptyset \neq 2$,$\emptyset \neq 3$and$\emptyset \neq \{6,7\}$2)$\{\}$could be a notation for$\emptyset$, it's unusual (= not used) and prone to missunderstanding, but it's logical. • But doesn't {6,7} contain$\emptyset$. If A is a subset of B then B contains everything present in A, so doesn't this mean everything in A belongs to B? – Uian Mail Mar 23 '16 at 13:15 • @UianMail :$\emptyset \neq \{ 6,7 \}$, and it's all that matter. You don't say that$3 \subset \{1,2, \{3,4\} \}$either – Tryss Mar 23 '16 at 13:18 • @UianMail You may be confused about how these symbols are pronounced.$\emptyset$is contained in$\{ 6,7 \}$; symbolically$\emptyset \subseteq \{ 6,7 \}$. But it is not an element of$\{ 6,7 \}$; symbolically$\emptyset \not \in \{ 6,7 \}$. The empty set can be an element of a set because it is an object in its own right. – Ian Mar 23 '16 at 13:41$\emptyset$is a subset of every set; not a member of every set.$\{\emptyset\}$is not empty; it has a member; namely$\emptyset$. By definition,$A\subseteq B$iff every element of$A$is contained in$B$. Is$\emptyset$an element of$\{1,2,3,\{6,7\}\}$? in other words: Is$\emptyset$one of these elements :$1,2,3,$or$\{6,7\}\$?

Here you can see good question and answer about using "{}" in set theory.