The LaTeX command for nexists (∄) is \nexists (there does not exist).
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Write \nexists inside math mode, that is between $…$ for inline math or in a \[ … \] or equation block for display math. This symbol needs the amssymb package, so add \usepackage{amssymb} to your preamble.
\nexists keeps its spacing inside a formula; for a word such as "and" in the middle of one, use \text{ and } from amsmath. With unicode-math under XeLaTeX or LuaLaTeX, typing ∄ in a formula gives the same symbol.
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Use \nexists in math mode after adding \usepackage{amssymb} to your preamble. For example, $p \nexists q$.
It comes from the amssymb package: add \usepackage{amssymb} to your preamble.