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  "body": "\n## Proof\n\nWe show that the two propositions have the same truth value for each set $X$ and each [[predicate|0T3DPBG981]] $P$ on $X$.\n\n1. Let $X$ be a set and let $P(x)$ be a predicate on $X$. For each $a \\in X$, the proposition $P(a)$ is true or false.\n2. By the definition of [[negation|0XR7A7RSXB]], $\\neg \\bigl( \\exists x \\in X, \\, P(x) \\bigr)$ is true exactly when $\\exists x \\in X, \\, P(x)$ is false.\n3. By the definition of the [[existential quantifier|0JW2FHW7G3]], $\\exists x \\in X, \\, P(x)$ is false exactly when there is no $a \\in X$ for which $P(a)$ is true.\n4. There is no such $a$ exactly when, for each $a \\in X$, $P(a)$ is not true. By step 1, this means that $P(a)$ is false for each $a \\in X$.\n5. By the definition of negation, $P(a)$ is false exactly when $\\neg P(a)$ is true. So step 4 says that $\\neg P(a)$ is true for each $a \\in X$.\n6. By the definition of the [[universal quantifier|0B88PQTJ66]], this is exactly when $\\forall x \\in X, \\, \\neg P(x)$ is true.\n7. By steps 2 to 6, the two propositions are true in the same cases, and false in the same cases. Hence $\\neg \\bigl( \\exists x \\in X, \\, P(x) \\bigr) \\equiv \\forall x \\in X, \\, \\neg P(x)$.\n\n## Idea\n\n\"Nothing has the property\" means \"each thing fails to have it\". The proof unfolds the meaning of the existential quantifier and of negation, one definition at a time.\n",
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