{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-set-membership",
  "semantic_family_id": "sf-set-membership",
  "type": "definition",
  "version": "0.8.0",
  "language": "zh-Hant",
  "titles": { "zh-Hant": "集合隸屬", "en": "Set membership" },
  "summary": { "zh-Hant": "符號 x∈A 表示元素 x 屬於集合 A。" },
  "statement": { "zh-Hant": "給定集合 A 與對象 x，x∈A 是一個命題，表示 x 是 A 的元素。" },
  "formula": {
    "tex": "x\\in A",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow></math>",
    "semantic_ast": {
      "type": "membership",
      "element": { "type": "symbol", "name": "x", "glyph": "x" },
      "set": { "type": "symbol", "name": "A", "glyph": "A" }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "fac41f5df7e2995de6d6b48a0cce188696ab53b3a4ebdbcc87c451009145c802"
    }
  },
  "assumptions": [
    { "id": "a1", "zh-Hant": "A 已被指定為一個集合。" }
  ],
  "symbols": [
    { "token": "x", "role": "element_candidate", "role_zh": "候選元素", "scope": "definition" },
    { "token": "A", "role": "set", "role_zh": "集合", "scope": "definition" },
    { "token": "∈", "role": "membership_relation", "role_zh": "隸屬關係", "scope": "definition" }
  ],
  "dependencies": [],
  "explanation": {
    "paragraphs_zh": [
      "集合像是一個由若干對象構成的數學容器，但集合本身不必依靠實際盒子或空間來存在。",
      "x∈A 只表示隸屬關係；它不表示 x 等於 A，也不表示 x 是 A 的子集合。",
      "在目前的 Lean 映射中，Set α 被表示為 α→Prop，因此 x∈A 與 A x 在定義上等價。"
    ],
    "common_misconception_zh": "混淆元素隸屬 x∈A 與子集合關係 B⊆A，或把程式容器成員測試視為所有集合論語義。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "formal_lean_set_semantics_available",
    "warning_zh": "Lean 已形式化 Set α 作為 α→Prop 時的隸屬語義；此結果不宣稱涵蓋所有集合論基礎，也不讓程式容器自動成為數學集合。",
    "evidence_refs": [
      {
        "id": "evidence-sha256-702de77019f352eb431eb67bd7b266491b7c36c922d952e44be2b11ba95bcb5d",
        "role": "formal_proof",
        "producer_id": "lean-mathlib"
      }
    ],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "active"
      }
    ]
  },
  "formalization": {
    "status": "formalized_lean_set_semantics",
    "target_systems": ["Lean 4", "Mathlib"],
    "next_obligation_zh": "擴充子集合、集合相等與基本集合運算物件，並保持隸屬、包含與等號三種關係分離。"
  },
  "provenance": {
    "origin": "evemisslab_original_mvp",
    "note_zh": "OCME 基礎數學知識物件；敘述為原創整理。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-07-31"
  }
}