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  "body": "\n## Statement\n\nLet $B$ be a [[Boolean algebra|06GKBTAA5E]] with operations $\\wedge$, $\\vee$, $\\neg$, least element $0$ and greatest element $1$.\nAn ideal of $B$ is a set $I \\subseteq B$ such that $0 \\in I$, $a \\vee b \\in I$ for all $a, b \\in I$, and $a \\in I$ whenever $a \\le b$ and $b \\in I$.\nAn ideal $I$ is prime if $1 \\notin I$ and, for each $a \\in B$, either $a \\in I$ or $\\neg a \\in I$.\nThen each ideal $I$ of $B$ with $1 \\notin I$ is contained in a prime ideal of $B$.\n\n## Remarks\n\nIn ZFC the proof applies [[Zorn's lemma|04RXTQA5ZA]] to the ideals that contain $I$ and omit $1$.\nA maximal one $P$ is prime: if $a \\notin P$ and $\\neg a \\notin P$, then the ideals generated by $P \\cup \\set{a}$ and by $P \\cup \\set{\\neg a}$ both contain $1$, and this forces $1 \\in P$.\nThe theorem is usually abbreviated BPI.\nBy duality, BPI is equivalent to the ultrafilter lemma: each filter on a set extends to an [[ultrafilter|SR56KCFYYS]].\nOver ZF, BPI follows from the [[axiom of choice|0HTGB13FE2]] but does not imply it.\nOver ZF, BPI is equivalent to the compactness theorem of first-order logic and to Stone's representation theorem for Boolean algebras.\nExample: in the power set Boolean algebra of a set $X$, with $x \\in X$, the family of subsets of $X$ that do not contain $x$ is a prime ideal.\n\n## Explanations\n\n### L1 human\n\nA Boolean algebra is an abstract version of the algebra of subsets of a set, with \"and\", \"or\" and \"not\".\nA prime ideal is a way to split the algebra into \"small\" elements and \"large\" elements, so that each element or its complement is small.\nThe theorem says that such a split always exists, extending any consistent choice of small elements.\nIt needs a weak form of the axiom of choice.\n\n### L2 human\n\nBPI is the choice principle behind many compactness arguments.\nIt gives the ultrafilter lemma, the compactness theorem of first-order logic, and the representation of every Boolean algebra as an algebra of clopen sets of a Stone space.\nIt also gives [[Tychonoff's theorem|0JTRSJKCFB]] for products of compact Hausdorff spaces, though not for arbitrary compact spaces, which needs full choice.\nSo it sits strictly between ZF and ZFC, and it is a common test of how much choice a theorem really needs.\n",
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