# Why is $\Bbb N$ $\not \subseteq \{\Bbb N\}$?

I am working through Hammack's Book of Proof to self-study proofs. This is also my first exposure to basic set theory. Example 1.6 lists 20 statements, all of which make sense to me except for one: $$\Bbb N\not \subseteq \{\Bbb N\}$$. The reason given is "because, for instance, $$1\in\Bbb N$$ but $$1 \not\in \{\Bbb N\}$$". I don't understand this because $$\Bbb N$$ contains $$1$$, so why doesn't the set of $$\Bbb N$$ also contain $$1$$?

I understand this is a very basic question, but I don't want to continue further in the book until I thoroughly understand each section.

• Subsets of $\{ \mathbb{N} \}$ are sets themselves. Think: $\mathbb{N}$ is a box, and the previous is a box containing that box. Commented Jun 10 at 16:22
• Since $x \in \left\{ y \right\} \iff x = y$ for any $x$ and $y$, $1 \ne \mathbb{N}$ implies $1 \notin \left\{ \mathbb{N} \right\}$. Commented Jun 10 at 16:22
• One little thing that may help: a subset of a set can't have 'more' elements than the set itself does (I'm being deliberately fuzzy here, but there is a true statement behind this). Now $\mathbb{N}$ has infinitely many elements, but $\{\mathbb{N}\}$ has only one... Commented Jun 10 at 16:24
• The only element of {$\mathbb N$} is the set $\mathbb N$ , nothing else , in particular not the elements of the set $\mathbb N$ Commented Jun 10 at 16:24
• Try to avoid the c word; it is too ambiguous to be helpful to your intuition. Instead of thinking "$\mathbb N$ contains $1$", think "$1$ is an element of $\mathbb N$". It should then become clear that "$1$ is not an element of $\{\mathbb N\}$". Commented Jun 10 at 16:36

$$\{\mathbb{N}\}$$ has only one element called $$\mathbb{N}$$.
Something like $$\{\{1,2,3,4,5\}\}$$. Can you claim $$1\in \{\{1,2,3,4,5\}\}=A$$?
No, since the set A has only one element called $$\{1,2,3,4,5\}$$.