TITLE: How to show $\{\{x\}\} \neq x$?
QUESTION [7 upvotes]: Let $x$ be any set. Using ZFC axioms, how to show $(x,x)=\{\{x\}\} \neq x$?
Similar question: $\bigcup x \neq x?$
Solved questions: $x \neq P(x)$ ($x$ is a subset of $x$ but $x \notin x$ by the Foundation axiom) 
$\{x\}=\{x,x\}\neq x$ (by the Foundation axiom)

REPLY [14 votes]: It seems the following: 
If $\{\{x\}\}=x$ then put $y=\{x\}$. Then $y\ni x\ni y\dots $,  that is impossible. 
$\bigcup\varnothing=\varnothing$.

REPLY [14 votes]: If $x = \{ \{ x \} \}$ then $x \in \{ x \} \in \{ \{ x \} \} = x$, and so the set $\{ x , \{ x \} \}$ has no $\in$-minimal element (contradicting Foundation).
The second statement is not true.  It is clearly not true for $\varnothing$, and as Andres Caicedo has mentioned below, it is not true for plenty of nonempty sets.