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  "body": "\n## Definition\n\nLet $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.\nThe proposition $\\exists! x \\in X, \\, P(x)$ is true when exactly one $a \\in X$ makes $P(a)$ true.\nIn terms of the other quantifiers, it is the proposition\n$$\\exists x \\in X, \\, \\bigl( P(x) \\land \\forall y \\in X, \\, (P(y) \\to y = x) \\bigr).$$\n\n## Remarks\n\nA proof of $\\exists! x, \\, P(x)$ usually has two parts: existence, an $x$ with $P(x)$, and uniqueness, if $P(x)$ and $P(y)$ then $x = y$.\nFor example, $\\exists! x \\in \\mathbb{R}, \\, 2x + 1 = 7$ is true, with $x = 3$.\n\n## Explanations\n\n### L1 human\n\n\"There is exactly one even prime\" says two things.\nThere is an even prime, namely $2$, and no other number is an even prime.\n",
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