"For no element" symbol The symbol $\forall$ is known to account every element on given set. Is there such symbol as "for no element", something like a crossed $\forall$?

"There is no $x$ such that $P(x)$" is the same with "(There is $x$ such that $P(x)$) does not hold." Hence Your quantifier is simply the negation of the existential quantifier $\exists$. We already have a symbol for this: $\nexists$. However, I rarely see this symbol in practice. (I can say that logicians, at least, use $\lnot\exists$ more than $\nexists$.)
Another way to state $\nexists x P(x)$, or instead, $\lnot\exists x P(x)$ is $\forall x \lnot P(x)$. Informally, it means "Every $x$ satisfies the negation of $P(x)$." It follows from de Morgan's law, and interestingly, the equivalence between $\lnot\exists xP(x)$ and $\forall x\lnot P(x)$ does not require the law of excluded middle. Hence the equivalence is constructively valid, unlike that between $\lnot\forall x P(x)$ and $\exists x \lnot P(x)$.
