LaTeX for all / exist symbols (uniqueness quantification, not exist)

Symbol Command
$\forall$ \forall
$\exist$ \exist
$\exists$ \exists
$\exists!$ \exists!
$\nexists$ \nexists

Examples

\forall x \in A \\~\\
\exist x \in A \\~\\
\exists x \in A \\~\\
\exists! x \in A

\[ \forall x \in A \\~\\ \exist x \in A \\~\\ \exists x \in A \\~\\ \exists! x \in A \]

Spacing after forall

\forall x \in A \\~\\
\forall \  x \in A \\~\\
\forall \, x \in A \\~\\
\forall \: x \in A \\~\\
\forall \; x \in A

\[ \forall x \in A \\~\\ \forall \ x \in A \\~\\ \forall \, x \in A \\~\\ \forall \: x \in A \\~\\ \forall \; x \in A \]

Spacing after exist

\exist x \in A \\~\\
\exist \  x \in A \\~\\
\exist \, x \in A \\~\\
\exist \: x \in A \\~\\
\exist \; x \in A

\[ \exist x \in A \\~\\ \exist \ x \in A \\~\\ \exist \, x \in A \\~\\ \exist \: x \in A \\~\\ \exist \; x \in A \]

LaTeX set

LaTeX