# Is there a formal proof concerning whether a set is (not) the subset of a set containing only that set?

I was wondering whether there is a formal proof that exists showing that some arbitrary set $$A$$ is not a subset of $$\{A\}$$ (though it would be an element)? It's not clear to me whether this is just a universally accepted axiom of set theory, or if there is a way to prove the relation (or lack thereof).

If it helps, I'd be happy with an answer in terms of the simple sets $$\{1\}$$ and $$\{\{1\}\}$$!

Edit: I had forgotten about the empty set! For my purposes, let's specifically exclude the case where $$A$$ = $$\emptyset$$.

• If $A$ is the empty set then it is a subset of $\{A\}.$ Jun 2, 2022 at 0:50
• @DavidK Conversely, if we have any $x \in A$, then we know that $A \notin A$ and therefore $x \neq A$. So $x \notin \{A\}$, and thus $A \nsubseteq \{A\}$. Jun 2, 2022 at 0:56
• For another start, notice $\{A\}$ has exactly two subsets: $\emptyset$ and $\{A\}$. Jun 2, 2022 at 1:03
• @MarkSaving Thanks for the response! I think my follow-up is essentially the same as my original question: while I think $A \notin A$ if $x \in A$ makes sense intuitively, is this something that can be formalized, or just taken for granted? Maybe I'm being a bit too dense here! Jun 2, 2022 at 1:19
• The fact that $A \notin A$ is a consequence of the Axiom of Regularity. Jun 2, 2022 at 1:37

By the Axiom of Regularity, every non-empty set $$A$$ contains an element that is disjoint from $$A$$.
Extending this axiom to the set containing non-empty $$A$$, denoted as $$\{A\}$$, there exists an element of $$\{A\}$$ that is disjoint from $$\{A\}$$. Since the only element of $$\{A\}$$ is $$A$$, it must follow that $$A$$ is disjoint from $$\{A\}$$ and $$A \cap \{A\} = \emptyset$$. Since $$A \in \{A\}$$, it must be that $$A \notin A$$.
Drawing from the above, assume $$x \neq \emptyset \in A$$. It follows that $$x \neq A$$ and therefore $$x \notin \{A\}$$. Thus, $$A \nsubseteq \{A\}$$.
• Except that if $A = \emptyset$ then $A \subseteq A$ is true. Jun 3, 2022 at 1:40