We start with a model of $\sf ZFC$, $V$, for simplicity we can imagine that $V$ satisfies $\sf GCH$ or even $V=L$.

Let $\Bbb P$ be $\operatorname{Add}(\omega,\omega_1)$, and let $G$ be a $V$-generic filter for $\Bbb P$. For $E\subseteq\omega_1$, we will write $G\restriction E$ as the restriction of $G$ to the reals with coordinates in $E$.

By the fact that the Cohen forcing is ccc, if $x\in\Bbb R^{V[G]}$, then there is some $E\in[\omega_1]^{\omega}\cap V$ such that $x\in V[G\restriction E]$. However, I was told there will be such $x$ without a minimal $E$.

What is an example for a real $x$ such that if $x\in V[G\restriction E]$, then there is some $E'\subsetneq E$ for which $x\in V[G\restriction E']$?