{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(P, \\le)$ be a partially ordered set and let $S$ be a subset of $P$.\nIf $b$ and $b'$ are both [[suprema|0BTMQ04556]] of $S$, then $b = b'$.\n\n## Remarks\n\nProof: $b$ is an upper bound and $b'$ is a least upper bound, thus $b' \\le b$. In the same way $b \\le b'$, and antisymmetry gives $b = b'$.\nThe same argument shows that a set has at most one infimum.\n\n## Explanations\n\n### L1 human\n\nA set cannot have two different least upper bounds.\nIf it had two, each one would be at most the other one, so they would be equal.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "06A06"
    ],
    "name": "Uniqueness of the supremum",
    "role": "proposition",
    "sources": [
      {
        "locator": "Existence and uniqueness",
        "url": "https://en.wikipedia.org/wiki/Infimum_and_supremum"
      }
    ],
    "tag": "04ETX30HT2",
    "uses": [
      "0BTMQ04556"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/04ETX30HT2/",
  "links": [
    {
      "field": "uses",
      "target": "0BTMQ04556"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/04ETX30HT2.md",
  "msc": [
    "06A06"
  ],
  "name": "Uniqueness of the supremum",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/04E/04ETX30HT2.md",
  "status": null,
  "tag": "04ETX30HT2",
  "uri": "m2://04ETX30HT2",
  "url": "https://math2.tamnd.com/wiki/uniqueness-of-the-supremum/",
  "writing": {
    "tag": "04ETX30HT2",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}