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      "title": "1. Propositional Logic (1.01-1.14)",
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      "index": 5,
      "name": "AxmPropConjIntro",
      "id": "AxmPropConjIntro",
      "legacy_id": "item-1.05",
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      "title": "A05",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 6,
      "name": "AxmPropDisjLeft",
      "id": "AxmPropDisjLeft",
      "legacy_id": "item-1.06",
      "num": "1.06",
      "title": "A06",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 7,
      "name": "AxmPropDisjRight",
      "id": "AxmPropDisjRight",
      "legacy_id": "item-1.07",
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      "title": "A07",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 8,
      "name": "AxmPropDisjElim",
      "id": "AxmPropDisjElim",
      "legacy_id": "item-1.08",
      "num": "1.08",
      "title": "A08",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛃\">𝛃</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛄\">𝛄</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/or-conn.html\" class=\"sym-link\" title=\"Logical disjunction connective (Constructor: HP (TpOr ...))\">∨</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛃\">𝛃</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛄\">𝛄</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "judgements_unicode_html": "<a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛃\">𝛃</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛄\">𝛄</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a>",
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      "side_condition_latex_linked": "",
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      "verification": "Postulated (Kernel Axiom)"
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    {
      "index": 9,
      "name": "AxmPropNegElim",
      "id": "AxmPropNegElim",
      "legacy_id": "item-1.09",
      "num": "1.09",
      "title": "A09",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "((¬⟦𝛂⟧)→(⟦𝛂⟧→⊥))",
      "prf_constructor": "PA9_prf",
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      "kernel_ast_unicode": "((¬⟦[𝛂@Fm.Expression]⟧)→(⟦[𝛂@Fm.Expression]⟧→⊥))",
      "formula_unicode_html": "<a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/neg-conn.html\" class=\"sym-link\" title=\"Logical negation connective (Constructor: HP (TpNeg ...))\">¬</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><span class=\"ansi-bold-blue\"><a href=\"symbols/bot-const.html\" class=\"sym-link\" title=\"Falsum / logical falsity constant (Constructor: HP TpBot)\">⊥</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/neg-conn.html}{\\neg} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\alpha}} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\alpha}} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/bot-const.html}{\\bot} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "[(𝛂:Fm)]",
      "judgements_unicode_html": "<a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a>",
      "judgements_latex_linked": "\\href{symbols/image-op.html}{[} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\alpha}} \\href{symbols/typing-colon.html}{:} \\href{symbols/fm-sort.html}{\\mathsf{Fm}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/image-op.html}{]} ",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Negation Elimination (Contradiction to Bottom): (¬⟦𝛂⟧) → (⟦𝛂⟧ → ⊥).",
      "verification": "Postulated (Kernel Axiom)"
    },
    {
      "index": 10,
      "name": "AxmPropNegIntro",
      "id": "AxmPropNegIntro",
      "legacy_id": "item-1.10",
      "num": "1.10",
      "title": "A10",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "verification": "Postulated (Kernel Axiom)"
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      "index": 11,
      "name": "AxmPropExFalso",
      "id": "AxmPropExFalso",
      "legacy_id": "item-1.11",
      "num": "1.11",
      "title": "A11",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "(⊥→⟦𝛂⟧)",
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      "kernel_ast_unicode": "(⊥→⟦[𝛂@Fm.Expression]⟧)",
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      "judgements_unicode_html": "<a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a>",
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      "side_condition_latex_linked": "",
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      "verification": "Postulated (Kernel Axiom)"
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      "index": 12,
      "name": "AxmPropExcludedMiddle",
      "id": "AxmPropExcludedMiddle",
      "legacy_id": "item-1.12",
      "num": "1.12",
      "title": "A12",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "(⟦𝛂⟧∨¬⟦𝛂⟧)",
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      "judgements_unicode_html": "<a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a>",
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      "index": 13,
      "name": "AxmPropVerum",
      "id": "AxmPropVerum",
      "legacy_id": "item-1.13",
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      "title": "A13",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
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      "side_condition_unicode_html": "",
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      "index": 14,
      "name": "RulModusPonens",
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      "legacy_id": "item-1.14",
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      "kind": "Inference Rule",
      "status": "Primitive / Derived Rule (Kernel)",
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      "name": "RulExistsAbstraction",
      "id": "RulExistsAbstraction",
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      "index": 20,
      "name": "RulGeneralisation",
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      "name": "DefEmptySet",
      "id": "DefEmptySet",
      "legacy_id": "item-3.04",
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      "title": "Empty set",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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      "name": "DefUniversalClass",
      "id": "DefUniversalClass",
      "legacy_id": "item-3.05",
      "num": "3.05",
      "title": "Universal class",
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      "status": "Definitional (Kernel AST)",
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      "index": 27,
      "name": "ThmEmptySetIsSubclass",
      "id": "ThmEmptySetIsSubclass",
      "legacy_id": "item-3.07",
      "num": "3.07",
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      "index": 28,
      "name": "AxmLeftInSound",
      "id": "AxmLeftInSound",
      "legacy_id": "item-3.08",
      "num": "3.08",
      "title": "“Membership respects equality from the left”",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable z (binder)\">z</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable z\">z</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" 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    {
      "index": 29,
      "name": "AxmRightInSound",
      "id": "AxmRightInSound",
      "legacy_id": "item-3.09",
      "num": "3.09",
      "title": "“Membership respects equality from the right”",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 32,
      "name": "AxmHeredity",
      "id": "AxmHeredity",
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      "num": "3.12",
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      "judgements_unicode_html": "",
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      "index": 33,
      "name": "AxmExponent",
      "id": "AxmExponent",
      "legacy_id": "item-3.13",
      "num": "3.13",
      "title": "“Exponent”(powerclass of a set is also a set)",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
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      "index": 34,
      "name": "AxmRegularity",
      "id": "AxmRegularity",
      "legacy_id": "item-3.14",
      "num": "3.14",
      "title": "“Regularity”(only universal class is equal to its powerclass)",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "(∀(X:Class)((𝒫(X)⊆X)→(V⊆X)))",
      "prf_constructor": "PEpsIndAx_prf",
      "scheme_of_types_unicode": "(∀([X@Variable]:[Class@Type])((𝒫([X@Class.Expression])⊆[X@Class.Expression])→(V⊆[X@Class.Expression])))",
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      "index": 35,
      "name": "DefTransitiveClass",
      "id": "DefTransitiveClass",
      "legacy_id": "item-3.15",
      "num": "3.15",
      "title": "Transitivity predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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href=\"symbols/iff-conn.html\" class=\"sym-link\" title=\"Logical equivalence connective (Constructor: HP (TpIff ...))\">⟷</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B (binder)\">B</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B\">B</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class 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application parenthesis\">(</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛙\">𝛙</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-white\"><a href=\"symbols/binder-brackets.html\" class=\"sym-link\" title=\"Left syntactic binder-variable bracket {| x |} (Constructor: IVariable_t)\">⦃</a></span><span class=\"ansi-light-yellow\"><a 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application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "index": 43,
      "name": "DefSimpleConditional",
      "id": "DefSimpleConditional",
      "legacy_id": "item-3.23",
      "num": "3.23",
      "title": "simple_if (polymorphic non-dependent conditional over Type)",
      "kind": "Definition",
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href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound type variable Tₑ\">Tₑ</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable t₁ (binder)\">t₁</a></span><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound type variable Tₑ\">Tₑ</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound branch term variable t₂ (binder)\">t₂</a></span><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound type variable Tₑ\">Tₑ</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/simple-if.html\" class=\"sym-link\" title=\"Polymorphic non-dependent conditional functional symbol (simple_if)\">simple_if</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound type variable Tₑ\">Tₑ</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/top-const.html\" class=\"sym-link\" title=\"Verum / logical truth constant (Constructor: HP TpTop)\">⊤</a><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable t₁\">t₁</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound branch term variable t₂\">t₂</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable t₁\">t₁</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/simple-if.html\" class=\"sym-link\" title=\"Polymorphic non-dependent conditional functional symbol (simple_if)\">simple_if</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound type variable Tₑ\">Tₑ</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/bot-const.html\" class=\"sym-link\" title=\"Falsum / logical falsity constant (Constructor: HP TpBot)\">⊥</a><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable t₁\">t₁</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound branch term variable t₂\">t₂</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound branch term variable t₂\">t₂</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{T_e} \\href{symbols/typing-colon.html}{:} \\href{symbols/type-sort.html}{\\mathsf{Type}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/comma-delim.html}{,}  \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/simple-if.html}{\\mathsf{simple\\_if}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/variables.html}{T_e} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/typing-colon.html}{:} \\href{symbols/bool-sort.html}{\\mathsf{Bool}} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/variables.html}{T_e} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/variables.html}{T_e} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/variables.html}{T_e} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/and-conn.html}{\\land} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{t_1} \\href{symbols/typing-colon.html}{:} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/variables.html}{T_e} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{t_2} \\href{symbols/typing-colon.html}{:} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/variables.html}{T_e} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/simple-if.html}{\\mathsf{simple\\_if}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/variables.html}{T_e} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/comma-delim.html}{,} \\href{symbols/top-const.html}{\\top} \\href{symbols/comma-delim.html}{,} \\href{symbols/variables.html}{t_1} \\href{symbols/comma-delim.html}{,} \\href{symbols/variables.html}{t_2} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/eq-pred.html}{=} \\href{symbols/variables.html}{t_1} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/and-conn.html}{\\land} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/simple-if.html}{\\mathsf{simple\\_if}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/variables.html}{T_e} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/comma-delim.html}{,} \\href{symbols/bot-const.html}{\\bot} \\href{symbols/comma-delim.html}{,} \\href{symbols/variables.html}{t_1} \\href{symbols/comma-delim.html}{,} \\href{symbols/variables.html}{t_2} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/eq-pred.html}{=} \\href{symbols/variables.html}{t_2} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Definitional (Kernel AST)"
    },
    {
      "index": 44,
      "name": "DefDependentConditional",
      "id": "DefDependentConditional",
      "legacy_id": "item-3.24",
      "num": "3.24",
      "title": "ifᵈₑₚ (dependent conditional over (Bool → Type))",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
      "formula_unicode": "∀(Tₑ:Bool→Type), ((ifᵈₑₚ:Π(b:Bool),(⟦Tₑ(⊤)⟧→(⟦Tₑ(⊥)⟧→⟦Tₑ(b)⟧)))∧∀(b:Bool)∀(t₁:⟦Tₑ(⊤)⟧)∀(t₂:⟦Tₑ(⊥)⟧)((ifᵈₑₚ(b,t₁,t₂):⟦Tₑ(b)⟧)∧((ifᵈₑₚ(⊤,t₁,t₂)=t₁)∧(ifᵈₑₚ(⊥,t₁,t₂)=t₂))))",
      "prf_constructor": "EIfDep",
      "scheme_of_types_unicode": "∀(Tₑ:Bool→Type), ((ifᵈₑₚ:Π(b:Bool),(⟦Tₑ(⊤)⟧→(⟦Tₑ(⊥)⟧→⟦Tₑ(b)⟧)))∧∀(b:Bool)∀(t₁:⟦Tₑ(⊤)⟧)∀(t₂:⟦Tₑ(⊥)⟧)((ifᵈₑₚ(b,t₁,t₂):⟦Tₑ(b)⟧)∧((ifᵈₑₚ(⊤,t₁,t₂)=t₁)∧(ifᵈₑₚ(⊥,t₁,t₂)=t₂))))",
      "kernel_ast_unicode": "∀(Tₑ:Bool→Type), ((ifᵈₑₚ:Π(b:Bool),(⟦Tₑ(⊤)⟧→(⟦Tₑ(⊥)⟧→⟦Tₑ(b)⟧)))∧∀(b:Bool)∀(t₁:⟦Tₑ(⊤)⟧)∀(t₂:⟦Tₑ(⊥)⟧)((ifᵈₑₚ(b,t₁,t₂):⟦Tₑ(b)⟧)∧((ifᵈₑₚ(⊤,t₁,t₂)=t₁)∧(ifᵈₑₚ(⊥,t₁,t₂)=t₂))))",
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(Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound type-family variable Tₑ\">Tₑ</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/bot-const.html\" class=\"sym-link\" title=\"Falsum / logical falsity constant (Constructor: HP TpBot)\">⊥</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation 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title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a 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      "name": "DefValueAtArg",
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      "index": 49,
      "name": "DefHasConcentratedValues",
      "id": "DefHasConcentratedValues",
      "legacy_id": "item-3.29",
      "num": "3.29",
      "title": "“Has concentrated set values” predicate",
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href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-green\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable 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title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "index": 50,
      "name": "DefFlatValues",
      "id": "DefFlatValues",
      "legacy_id": "item-3.30",
      "num": "3.30",
      "title": "“Has flat proper values” predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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      "index": 51,
      "name": "DefFamily",
      "id": "DefFamily",
      "legacy_id": "item-3.31",
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      "title": "“Is a family” predicate",
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      "status": "Definitional (Kernel AST)",
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      "prf_constructor": "TpFamily",
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      "index": 52,
      "name": "DefClassFunction",
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      "legacy_id": "item-3.32",
      "num": "3.32",
      "title": "“Is a class-function” predicate",
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    {
      "index": 53,
      "name": "DefInjectiveFunction",
      "id": "DefInjectiveFunction",
      "legacy_id": "item-3.33",
      "num": "3.33",
      "title": "“Is injective” predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable F\">F</a></span><span class=\"ansi-light-magenta\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable y\">y</a></span><span class=\"ansi-light-magenta\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable x\">x</a></span><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable y\">y</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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    {
      "index": 54,
      "name": "DefSetFunction",
      "id": "DefSetFunction",
      "legacy_id": "item-3.34",
      "num": "3.34",
      "title": "“Is a function” predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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      "name": "DefSemiSingleton",
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      "num": "3.35",
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      "index": 56,
      "name": "DefSemiSingletonSubclass",
      "id": "DefSemiSingletonSubclass",
      "legacy_id": "item-3.36",
      "num": "3.36",
      "title": "“Semi-singleton subset” relation (s sss A)",
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      "index": 58,
      "name": "DefSigmaModel",
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      "legacy_id": "item-3.38",
      "num": "3.38",
      "title": "“Dependent Sum (Sigma-Type)” model over a family",
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      "judgements_latex_linked": "",
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      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Explicit typed term interpretation / valuation (μ_A : A.Expression → Class.Expression) of the dependent sum Σ(x:Dom(F)) over a family F in the fixed background model.",
      "verification": "Definitional (Kernel AST)"
    },
    {
      "index": 59,
      "name": "DefPiPredicateModel",
      "id": "DefPiPredicateModel",
      "legacy_id": "item-3.39",
      "num": "3.39",
      "title": "“Dependent Product (Pi-Type)” predicate and section model",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "(∀(B:Class)∀(f:Class)(((B family)∧((f family)∧(Dom(f)=Dom(B))))→∀(x:Class)((x∈Dom(B))→(f(x)∈B(x)))))",
      "prf_constructor": "CpPi",
      "scheme_of_types_unicode": "(∀([B@Variable]:[Class@Type])∀([f@Variable]:[Class@Type])((([B@Class.Expression] family)∧(([f@Class.Expression] family)∧(Dom([f@Class.Expression])=Dom([B@Class.Expression]))))→∀([x@Variable]:[Class@Type])(([x@Class.Expression]∈Dom([B@Class.Expression]))→([f@Class.Expression]([x@Class.Expression])∈[B@Class.Expression]([x@Class.Expression])))))",
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      "index": 63,
      "name": "AxmSizeLimitation",
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      "legacy_id": "item-3.44",
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      "index": 67,
      "name": "DefTightClass",
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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/numeral-0.html}{0} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)}  \\href{symbols/eq-pred.html}{=}  \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/delimiters-brackets.html}{)}  \\href{symbols/and-conn.html}{\\land}  \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/numeral-1.html}{1} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)}  \\href{symbols/eq-pred.html}{=}  \\href{symbols/singleton-op.html}{\\{} \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/singleton-op.html}{\\}} \\href{symbols/delimiters-brackets.html}{)}  \\href{symbols/and-conn.html}{\\land}  \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/numeral-2.html}{2} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)}  \\href{symbols/eq-pred.html}{=}  \\href{symbols/upair-op.html}{\\{} \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/comma-delim.html}{,} \\href{symbols/singleton-op.html}{\\{} \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/singleton-op.html}{\\}} \\href{symbols/upair-op.html}{\\}} \\href{symbols/delimiters-brackets.html}{)}  \\href{symbols/and-conn.html}{\\land}  \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/numeral-3.html}{3} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)}  \\href{symbols/eq-pred.html}{=}  \\href{symbols/upair-op.html}{\\{} \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/comma-delim.html}{,} \\href{symbols/singleton-op.html}{\\{} \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/singleton-op.html}{\\}} \\href{symbols/comma-delim.html}{,} \\href{symbols/upair-op.html}{\\{} \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/comma-delim.html}{,} \\href{symbols/singleton-op.html}{\\{} \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/singleton-op.html}{\\}} \\href{symbols/upair-op.html}{\\}} \\href{symbols/upair-op.html}{\\}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Explicit typed term interpretation / valuation (μ_Nat : Nat.Expression → Class.Expression) mapping numeral terms 0, 1, 2, 3 into their von Neumann ordinal classes in the fixed background model.",
      "verification": "Definitional (Kernel AST)"
    },
    {
      "index": 71,
      "name": "DefClassLists",
      "id": "DefClassLists",
      "legacy_id": "item-3.51",
      "num": "3.51",
      "title": "General definition of lists of classes which applicable not only for sets, but for proper classes too(TBI)",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
      "formula_unicode": "(mark(A,n) = ({{0,n}} ∪ {({{{a}},n}:Class) | (a ∈ A)})) ∧ (([] = ∅) ∧ (([A] = mark(A,1)) ∧ (([A,B] = (mark(A,1) ∪ mark(B,2))) ∧ (((A,B) = [A,B]) ∧ ((pr1(L) = {a | ({{{a}},1} ∈ L)}) ∧ (pr2(L) = {b | ({{{b}},2} ∈ L)}))))))",
      "prf_constructor": "CpOPair",
      "scheme_of_types_unicode": "(mark(A,n) = ({{0,n}} ∪ {({{{a}},n}:Class) | (a ∈ A)})) ∧ (([] = ∅) ∧ (([A] = mark(A,1)) ∧ (([A,B] = (mark(A,1) ∪ mark(B,2))) ∧ (((A,B) = [A,B]) ∧ ((pr1(L) = {a | ({{{a}},1} ∈ L)}) ∧ (pr2(L) = {b | ({{{b}},2} ∈ L)}))))))",
      "kernel_ast_unicode": "(mark(A,n) = ({{0,n}} ∪ {({{{a}},n}:Class) | (a ∈ A)})) ∧ (([] = ∅) ∧ (([A] = mark(A,1)) ∧ (([A,B] = (mark(A,1) ∪ mark(B,2))) ∧ (((A,B) = [A,B]) ∧ ((pr1(L) = {a | ({{{a}},1} ∈ L)}) ∧ (pr2(L) = {b | ({{{b}},2} ∈ L)}))))))",
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href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free class variable A\">A</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension { x | ϗ } (AST constructor: CpKappa, binder)\">}</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a> <a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a 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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free class variable A\">A</a></span><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a> <a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a> <a href=\"symbols/mark-op.html\" class=\"sym-link\" title=\"Dunaev class-list mark functional symbol (mark)\">mark</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free class variable A\">A</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/numeral-1.html\" class=\"sym-link\" title=\"von Neumann natural number 1 = {0}\">1</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a> <a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / 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      "name": "DefKuratowskiPair",
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      "index": 76,
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      "index": 77,
      "name": "AxmOmegaIsSet",
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      "legacy_id": "item-3.57",
      "num": "3.57",
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      "index": 78,
      "name": "ThmUniversalClassInductive",
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      "index": 79,
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      "formula_unicode_html": "<span class=\"ansi-bold-blue\"><a href=\"symbols/ind-pred.html\" class=\"sym-link\" title=\"Is successor-inductive (Peano-inductive) unary predicate symbol (Constructor: HP (TpInd ...))\">Ind</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/on-const.html\" class=\"sym-link\" title=\"Class of all ordinal sets On constant (Constructor: HC CpOn)\">On</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 81,
      "name": "ThmOmegaTransitive",
      "id": "ThmOmegaTransitive",
      "legacy_id": "item-3.61",
      "num": "3.61",
      "title": "ω is transitive",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Tr(ω)",
      "prf_constructor": "TpTrans",
      "scheme_of_types_unicode": "Tr(ω)",
      "kernel_ast_unicode": "Tr(ω)",
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      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 82,
      "name": "ThmOmegaOrdinal",
      "id": "ThmOmegaOrdinal",
      "legacy_id": "item-3.62",
      "num": "3.62",
      "title": "ω is ordinal",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Ord(ω)",
      "prf_constructor": "TpOrd",
      "scheme_of_types_unicode": "Ord(ω)",
      "kernel_ast_unicode": "Ord(ω)",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
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      "index": 83,
      "name": "ThmOmegaInOrdinals",
      "id": "ThmOmegaInOrdinals",
      "legacy_id": "item-3.63",
      "num": "3.63",
      "title": "ω is in the class of all ordinal numbers(=sets)",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
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      "kernel_ast_unicode": "(ω∈On)",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
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      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
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    {
      "index": 84,
      "name": "DefIsCardinal",
      "id": "DefIsCardinal",
      "legacy_id": "item-3.64",
      "num": "3.64",
      "title": "“Is cardinal” predicate",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
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      "prf_constructor": "TpCard",
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      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
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    {
      "index": 85,
      "name": "ThmOmegaCardinal",
      "id": "ThmOmegaCardinal",
      "legacy_id": "item-3.65",
      "num": "3.65",
      "title": "ω is cardinal",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Card(ω)",
      "prf_constructor": "TpCard",
      "scheme_of_types_unicode": "Card(ω)",
      "kernel_ast_unicode": "Card(ω)",
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      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 86,
      "name": "ThmOrdinalsCardinal",
      "id": "ThmOrdinalsCardinal",
      "legacy_id": "item-3.66",
      "num": "3.66",
      "title": "On is a cardinal class",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Card(On)",
      "prf_constructor": "TpCard",
      "scheme_of_types_unicode": "Card(On)",
      "kernel_ast_unicode": "Card(On)",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
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      "verification": "Verified (Kernel)"
    },
    {
      "index": 87,
      "name": "ThmUniversalClassNotCardinal",
      "id": "ThmUniversalClassNotCardinal",
      "legacy_id": "item-3.67",
      "num": "3.67",
      "title": "Class of all sets is not a cardinal class",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "(¬Card(V))",
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      "kernel_ast_unicode": "(¬Card(V))",
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      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/neg-conn.html}{\\neg} \\href{symbols/card-pred.html}{\\mathrm{Card}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/universal-class-const.html}{\\mathbf{V}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
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    {
      "index": 88,
      "name": "DefGrothendieckUniverse",
      "id": "DefGrothendieckUniverse",
      "legacy_id": "item-3.68",
      "num": "3.68",
      "title": "“Is a universe” predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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href=\"symbols/iff-conn.html\" class=\"sym-link\" title=\"Logical equivalence connective (Constructor: HP (TpIff ...))\">⟷</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/omega-const.html\" class=\"sym-link\" title=\"Smallest successor-inductive ordinal and subseteq-minimal successor-inductive class (Constructor: HC CpOmega)\">ω</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-blue\"><a href=\"symbols/trans-pred.html\" class=\"sym-link\" title=\"Transitivity unary predicate symbol (Constructor: HP (TpTrans ...))\">Tr</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><span class=\"ansi-bold-blue\"><a href=\"symbols/hereditary-pred.html\" class=\"sym-link\" title=\"Is hereditary unary predicate symbol (Constructor: HP (TpHereditary ...))\">Hereditary</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-blue\"><a href=\"symbols/powerful-pred.html\" class=\"sym-link\" title=\"Is powerful unary predicate symbol (Constructor: HP (TpPowerful ...))\">Powerful</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><span class=\"ansi-bold-blue\"><a href=\"symbols/tight-pred.html\" class=\"sym-link\" title=\"Is tight unary predicate symbol (Constructor: HP (TpTight ...))\">Tight</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered 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title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "judgements_latex_linked": "",
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      "side_condition_latex_linked": "",
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      "verification": "Definitional (Kernel AST)"
    },
    {
      "index": 89,
      "name": "ThmUniversalClassIsUniverse",
      "id": "ThmUniversalClassIsUniverse",
      "legacy_id": "item-3.69",
      "num": "3.69",
      "title": "The class of all sets is a universe",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "(V universe)",
      "prf_constructor": "PVUniverseAx_prf",
      "scheme_of_types_unicode": "(V universe)",
      "kernel_ast_unicode": "(V universe)",
      "formula_unicode_html": "<a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/universal-class-const.html\" class=\"sym-link\" title=\"Universal class V constant (Constructor: HC CpSets)\">V</a></span> <a href=\"symbols/universe-pred.html\" class=\"sym-link\" title=\"Is a Grothendieck/Tarski universe unary predicate symbol (Constructor: HP (TpUniverse ...))\">universe</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/universal-class-const.html}{\\mathbf{V}}  \\href{symbols/universe-pred.html}{\\mathsf{universe}} \\href{symbols/delimiters-brackets.html}{)} ",
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      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Formal kernel proof (v_universe_thm_proof : unit -> Thm, verified by Thm.checkPipeline and Thm.slow_check) deriving (𝘍, [] ⊢ (V universe)) from 3.68 #DefGrothendieckUniverse (PKernelDef_prf) and the 6 component lemmas: (1) (ω ∈ V) via 3.57 #AxmOmegaIsSet (POmegaSetAx_prf), 3.05 #DefUniversalClass (PKernelDef_prf), 3.08 #AxmLeftInSound (PLeftInSoundAx_prf), and 3.01 #RulClassComprehension (PComprAx_prf); (2) (V transitive) via 3.15 #DefTransitiveClass (PKernelDef_prf), 3.06 #DefSubclass (PKernelDef_prf), and 3.02 #DefIsSet (PSetDef_prf); (3) (V Hereditary) via 3.44 #DefHereditaryClass (PKernelDef_prf) and 3.12 #AxmHeredity (PHeredityAx_prf); (4) (V Powerful) via 3.45 #DefPowerfulClass (PKernelDef_prf) and 3.13 #AxmExponent (PPowSetAx_prf); (5) (V Tight) via 3.47 #DefTightClass (PKernelDef_prf), 3.43 #AxmSizeLimitation (PSizeLimAx_prf), and 3.41/3.42 #DefCardinality/#DefCardLess (PCardLessOrderAx_prf); (6) (V Pairful) via 3.46 #DefPairfulClass (PKernelDef_prf) and 3.49 #DefUnorderedPair sethood (PPairSetAx_prf). Backup axiom constructor PVUniverseAx_prf (v_universe_axm_backup_entry) is also retained in the kernel just in case.",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 90,
      "name": "AxmTarskiUniverses",
      "id": "AxmTarskiUniverses",
      "legacy_id": "item-3.70",
      "num": "3.70",
      "title": "“Tarski's axiom”",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "(∀(x:Class)((x set)→∃(u:Class)(((u universe)∧(x∈u))∧(u set))))",
      "prf_constructor": "TpUniverse",
      "scheme_of_types_unicode": "(∀([x@Variable]:[Class@Type])(([x@Class.Expression] set)→∃([u@Variable]:[Class@Type])((([u@Class.Expression] universe)∧([x@Class.Expression]∈[u@Class.Expression]))∧([u@Class.Expression] set))))",
      "kernel_ast_unicode": "(∀([x@Variable]:[Class@Type])(([x@Class.Expression] set)→∃([u@Variable]:[Class@Type])((([u@Class.Expression] universe)∧([x@Class.Expression]∈[u@Class.Expression]))∧([u@Class.Expression] set))))",
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on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable x\">x</a></span> <a href=\"symbols/set-pred.html\" class=\"sym-link\" title=\"Is a set unary predicate symbol (Constructor: HP (TpSet ...))\">set</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/exists-binder.html\" class=\"sym-link\" title=\"Existential quantifier binder (Constructor: HP (TpExis ...))\">∃</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable u (binder)\">u</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable u\">u</a></span> <a href=\"symbols/universe-pred.html\" class=\"sym-link\" title=\"Is a Grothendieck/Tarski universe unary predicate symbol (Constructor: HP (TpUniverse ...))\">universe</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable x\">x</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable u\">u</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable u\">u</a></span> <a href=\"symbols/set-pred.html\" class=\"sym-link\" title=\"Is a set unary predicate symbol (Constructor: HP (TpSet ...))\">set</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{x} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{x}  \\href{symbols/set-pred.html}{\\mathsf{set}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/exists-binder.html}{\\exists} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{u} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{u}  \\href{symbols/universe-pred.html}{\\mathsf{universe}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/and-conn.html}{\\land} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{x} \\href{symbols/in-pred.html}{\\in} \\href{symbols/variables.html}{u} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/and-conn.html}{\\land} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{u}  \\href{symbols/set-pred.html}{\\mathsf{set}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
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      "judgements_unicode_html": "",
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      "verification": "Postulated (Kernel Axiom)"
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    {
      "index": 91,
      "name": "ThmTransitiveClassInduction",
      "id": "ThmTransitiveClassInduction",
      "legacy_id": "item-3.71",
      "num": "3.71",
      "title": "General form of the transfinite ∈-induction principle on a transitive class",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "(∀(K:Class)∀(A:Class)((Tr(K)∧((K∩𝒫(A))⊆A))→(K⊆A)))",
      "prf_constructor": "PEpsIndAx_prf",
      "scheme_of_types_unicode": "(∀([K@Variable]:[Class@Type])∀([A@Variable]:[Class@Type])((Tr([K@Class.Expression])∧(([K@Class.Expression]∩𝒫([A@Class.Expression]))⊆[A@Class.Expression]))→([K@Class.Expression]⊆[A@Class.Expression])))",
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class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "index": 92,
      "name": "AxmSoundnessOfBaseTheory",
      "id": "AxmSoundnessOfBaseTheory",
      "legacy_id": "item-3.72",
      "num": "3.72",
      "title": "Soundness of the theory 𝘛",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 93,
      "name": "AxmSoundnessOfFullTheory",
      "id": "AxmSoundnessOfFullTheory",
      "legacy_id": "item-3.73",
      "num": "3.73",
      "title": "Soundness of the theory 𝘍",
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      "status": "Postulated (Kernel Axiom)",
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      "legacy_id": "item-3.74",
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      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/formula-metavariables.html}{\\Gamma} \\href{symbols/typing-colon.html}{:} \\href{symbols/ctx-sort.html}{\\mathsf{Ctx}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/typing-colon.html}{:} \\href{symbols/fm-sort.html}{\\mathsf{Fm}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/formula-metavariables.html}{\\Gamma} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-fau.html}{\\mathcal{F}} \\href{symbols/comma-delim.html}{,} \\href{symbols/formula-metavariables.html}{\\Gamma} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Theory extension monotonicity (𝘛 ⊆ 𝘍): every theorem of the r.e. base theory 𝘛 is also a theorem of its non-r.e. Henkin completion 𝘍.",
      "verification": "Postulated (Kernel Axiom)"
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    {
      "index": 96,
      "name": "DefDunaevOrderedPair",
      "id": "DefDunaevOrderedPair",
      "legacy_id": "item-3.76",
      "num": "3.76",
      "title": "Ordered pair(Dunaev's tagged class pair)",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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title=\"Bound class variable B\">B</a></span><span class=\"ansi-light-green\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-green\"><a href=\"symbols/singleton-op.html\" class=\"sym-link\" title=\"Singleton set {a} (AST constructor: CpSingleton, functional symbol)\">{</a></span><span class=\"ansi-light-green\"><a href=\"symbols/upair-op.html\" class=\"sym-link\" title=\"Unordered pair {a, b} (AST constructor: CpUPair, 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class=\"sym-link\" title=\"Unordered pair {a, b} (AST constructor: CpUPair, functional symbol)\">}</a></span><span class=\"ansi-light-green\"><a href=\"symbols/singleton-op.html\" class=\"sym-link\" title=\"Singleton set {a} (AST constructor: CpSingleton, functional symbol)\">}</a></span><a href=\"symbols/binunion-op.html\" class=\"sym-link\" title=\"Binary union functional symbol (Constructor: HC (CpBinUnion ...))\">∪</a><span class=\"ansi-light-red\"><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension { x | ϗ } (AST constructor: CpKappa, binder)\">{</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable x (binder)\">x</a></span><span class=\"ansi-light-red\"><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension separator | in { x | ϗ } (AST constructor: CpKappa, binder)\">|</a></span><a href=\"symbols/exists-binder.html\" 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application parenthesis\">)</a><span class=\"ansi-light-red\"><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension { x | ϗ } (AST constructor: CpKappa, binder)\">}</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/binunion-op.html\" class=\"sym-link\" title=\"Binary union functional symbol (Constructor: HC (CpBinUnion ...))\">∪</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-green\"><a href=\"symbols/singleton-op.html\" class=\"sym-link\" title=\"Singleton set {a} (AST constructor: CpSingleton, functional symbol)\">{</a></span><span class=\"ansi-light-green\"><a href=\"symbols/upair-op.html\" class=\"sym-link\" title=\"Unordered pair {a, b} (AST constructor: CpUPair, functional 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conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-red\"><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension { x | ϗ } (AST constructor: CpKappa, binder)\">{</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable b (binder)\">b</a></span><span class=\"ansi-light-red\"><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension separator | in { x | ϗ } (AST constructor: CpKappa, binder)\">|</a></span><span class=\"ansi-light-green\"><a href=\"symbols/upair-op.html\" 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href=\"symbols/singleton-op.html\" class=\"sym-link\" title=\"Singleton set {a} (AST constructor: CpSingleton, functional symbol)\">}</a></span><span class=\"ansi-light-green\"><a href=\"symbols/singleton-op.html\" class=\"sym-link\" title=\"Singleton set {a} (AST constructor: CpSingleton, functional symbol)\">}</a></span><span class=\"ansi-light-green\"><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a></span><span class=\"ansi-light-green\"><a href=\"symbols/upair-op.html\" class=\"sym-link\" title=\"Unordered pair {a, b} (AST constructor: CpUPair, functional symbol)\">{</a></span><span class=\"ansi-bold-yellow\"><a href=\"symbols/emptyset-const.html\" class=\"sym-link\" title=\"Empty set / empty class constant (Constructor: HC CpEmptySet)\">∅</a></span><span class=\"ansi-light-green\"><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a></span><span class=\"ansi-light-green\"><a href=\"symbols/singleton-op.html\" class=\"sym-link\" title=\"Singleton set {a} (AST constructor: CpSingleton, functional symbol)\">{</a></span><span class=\"ansi-bold-yellow\"><a href=\"symbols/emptyset-const.html\" class=\"sym-link\" title=\"Empty set / empty class constant (Constructor: HC CpEmptySet)\">∅</a></span><span class=\"ansi-light-green\"><a href=\"symbols/singleton-op.html\" class=\"sym-link\" title=\"Singleton set {a} (AST constructor: CpSingleton, functional symbol)\">}</a></span><span class=\"ansi-light-green\"><a href=\"symbols/upair-op.html\" class=\"sym-link\" title=\"Unordered pair {a, b} (AST constructor: CpUPair, functional symbol)\">}</a></span><span class=\"ansi-light-green\"><a href=\"symbols/upair-op.html\" class=\"sym-link\" title=\"Unordered pair {a, b} (AST constructor: CpUPair, functional symbol)\">}</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-green\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable A\">A</a></span><span class=\"ansi-light-green\"><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B\">B</a></span><span class=\"ansi-light-green\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><span class=\"ansi-light-red\"><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension { x | ϗ } (AST constructor: CpKappa, binder)\">}</a></span><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B\">B</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{A} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{B} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/class-comprehension.html}{\\{} \\href{symbols/variables.html}{b} \\href{symbols/class-comprehension.html}{ \\mid } \\href{symbols/upair-op.html}{\\{} \\href{symbols/singleton-op.html}{\\{} \\href{symbols/singleton-op.html}{\\{} \\href{symbols/singleton-op.html}{\\{} \\href{symbols/variables.html}{b} \\href{symbols/singleton-op.html}{\\}} \\href{symbols/singleton-op.html}{\\}} \\href{symbols/singleton-op.html}{\\}} \\href{symbols/comma-delim.html}{,} \\href{symbols/upair-op.html}{\\{} \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/comma-delim.html}{,} \\href{symbols/singleton-op.html}{\\{} \\href{symbols/emptyset-const.html}{\\emptyset} \\href{symbols/singleton-op.html}{\\}} \\href{symbols/upair-op.html}{\\}} \\href{symbols/upair-op.html}{\\}} \\href{symbols/in-pred.html}{\\in} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{A} \\href{symbols/comma-delim.html}{,} \\href{symbols/variables.html}{B} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/class-comprehension.html}{\\}} \\href{symbols/eq-pred.html}{=} \\href{symbols/variables.html}{B} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Recovers B from (A,B) via tag 2 without requiring A or B to be sets.",
      "verification": "Verified (Kernel)"
    }
      ]
    },
    {
      "id": "sec-4",
      "title": "4. Metalogical Laws (4.01-4.06)",
      "range": "4.01-4.06",
      "description": "Explicit substitution, context encoding, co-necessitation, uniqueness quantifiers, and reflected comprehension.",
      "entries": [
    {
      "index": 99,
      "name": "DefMetaInstantiation",
      "id": "DefMetaInstantiation",
      "legacy_id": "item-4.01",
      "num": "4.01",
      "title": "Instantiation(meta,WIP)",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "(∀(A:Type)∀(G:Ctx.Expression)∀(𝛗:Fm)∀(t:eval(A).Expression)∀(v:Variable)(𝘞,⟦G⟧⊢‹(∀(⦃v⦄:⟦A⟧)⟦𝛗⟧)→(let ⟦v⟧ = ⟦t⟧ in ⟦𝛗⟧)›))",
      "prf_constructor": "ELet",
      "scheme_of_types_unicode": "(∀([A@Variable]:[Type@Type])∀([G@Variable]:[Ctx.Expression@Type])∀([𝛗@Variable]:[Fm@Type])∀([t@Variable]:[eval([A@Type.Expression.Expression]).Expression@Type])∀([v@Variable]:[Variable@Type])(𝘞,⟦[G@Ctx.Expression.Expression]⟧⊢‹(∀([⦃[v@Variable.Expression]⦄@Variable]:[⟦[A@Type.Expression.Expression]⟧@Type])⟦[𝛗@Fm.Expression]⟧)→(let ⟦[v@Name.Expression]⟧ = ⟦[t@Class.Expression.Expression]⟧ in ⟦[𝛗@Fm.Expression]⟧)›))",
      "kernel_ast_unicode": "(∀([A@Variable]:[Type@Type])∀([G@Variable]:[Ctx.Expression@Type])∀([𝛗@Variable]:[Fm@Type])∀([t@Variable]:[eval([A@Type.Expression.Expression]).Expression@Type])∀([v@Variable]:[Variable@Type])(𝘞,⟦[G@Ctx.Expression.Expression]⟧⊢‹(∀([⦃[v@Variable.Expression]⦄@Variable]:[⟦[A@Type.Expression.Expression]⟧@Type])⟦[𝛗@Fm.Expression]⟧)→(let ⟦[v@Name.Expression]⟧ = ⟦[t@Class.Expression.Expression]⟧ in ⟦[𝛗@Fm.Expression]⟧)›))",
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(Ctx)\">Ctx</a></span><a href=\"symbols/dot-delim.html\" class=\"sym-link\" title=\"Lambda binder body and sort projection separator (.)\">.</a><span class=\"ansi-light-green\"><a href=\"symbols/expression-sort.html\" class=\"sym-link\" title=\"Parameterized syntactic AST expression sort constructor (A.Expression)\">Expression</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛗 (binder)\">𝛗</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-light-green\"><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound term expression variable t (binder)\">t</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><a href=\"symbols/eval.html\" class=\"sym-link\" title=\"Metatheoretic formula valuation / truth evaluation (eval)\">eval</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free type variable A\">A</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/dot-delim.html\" class=\"sym-link\" title=\"Lambda binder body and sort projection separator (.)\">.</a><span class=\"ansi-light-green\"><a href=\"symbols/expression-sort.html\" class=\"sym-link\" title=\"Parameterized syntactic AST expression sort constructor (A.Expression)\">Expression</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable v (binder)\">v</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-light-green\"><a href=\"symbols/variable-sort.html\" class=\"sym-link\" title=\"Syntactic variable identifier sort (Variable)\">Variable</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-wau.html\" class=\"sym-link\" title=\"Free theory variable 𝘞\">𝘞</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound context variable G\">G</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-white\"><a href=\"symbols/binder-brackets.html\" class=\"sym-link\" title=\"Left syntactic binder-variable bracket {| x |} (Constructor: IVariable_t)\">⦃</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable v\">v</a></span><span class=\"ansi-white\"><a href=\"symbols/binder-brackets.html\" class=\"sym-link\" title=\"Right syntactic binder-variable bracket {| x |} (Constructor: IVariable_t)\">⦄</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free type variable A\">A</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛗\">𝛗</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/substitution-let.html\" class=\"sym-link\" title=\"Explicit local substitution binder (Constructor: ESubst / HSubst)\">let</a> <a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free variable-name variable v\">v</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a> <a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a> <a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free term expression variable t\">t</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a> <a href=\"symbols/substitution-let.html\" class=\"sym-link\" title=\"Explicit local substitution scope (Constructor: ESubst / HSubst)\">in</a> <a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛗\">𝛗</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "name": "DefMetaGeneralisation",
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      "verification": "Postulated (Kernel Axiom)"
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      "index": 101,
      "name": "DefMetaConecessitation",
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      "num": "4.03",
      "title": "CoNecessitation(meta,WIP)",
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      "status": "Primitive / Derived Rule (Kernel)",
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      "comment": "If a formula  is derived in 𝘍 from the empty context, then the formula 𝛗 is derived in 𝘍 from the context Γ. Why this holds for 𝘍: for an arbitrary theory 𝘞, concluding (𝘞, Γ⊢𝛗) from (𝘞, [] ⊢ ) requires 1-consistency / Σ₁-soundness of 𝘞 (and for the r.e. base theory 𝘛, asserting internal co-necessitation would by Löb's theorem 5.01 trivialize 𝘛). Because 𝘍 is a non-r.e. sound Henkin extension equipped with Soundness 3.73 ((𝘍, [] ⊢ ψ) → ⟦ψ⟧), instantiating 3.73 at ψ :=  directly yields (𝘍, Γ⊢𝛗).",
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      "name": "DefAtMostOneBinder",
      "id": "DefAtMostOneBinder",
      "legacy_id": "item-4.04",
      "num": "4.04",
      "title": "Quantifier “unique”(meta,WIP)",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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    {
      "index": 103,
      "name": "DefExistsUniqueBinder",
      "id": "DefExistsUniqueBinder",
      "legacy_id": "item-4.05",
      "num": "4.05",
      "title": "Quantifier “exists and unique”(meta,WIP)",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
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      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
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    {
      "index": 104,
      "name": "AxmMetaClassComprehension",
      "id": "AxmMetaClassComprehension",
      "legacy_id": "item-4.06",
      "num": "4.06",
      "title": "Class comprehension axiom schema(meta,WIP)",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "prf_constructor": "PComprAx_prf",
      "scheme_of_types_unicode": "(∀([𝛗@Variable]:[Fm@Type])∀([x@Variable]:[Name@Type])∀([k@Variable]:[Class.Expression@Type])(Fm.FFI(([k@Class.Expression.Expression]:Class.Expression),‹[x@Name.Expression]›,[𝛗@Fm.Expression])→(𝘛,[]⊢‹(⟦[k@Class.Expression.Expression]⟧∈{[⦃[x@Variable.Expression]⦄@Class.Expression]|⟦[𝛗@Fm.Expression]⟧})⟷((⟦[k@Class.Expression.Expression]⟧ set)∧(let [x@Name.Expression] = ⟦[k@Class.Expression.Expression]⟧ in ⟦[𝛗@Fm.Expression]⟧))›)))",
      "kernel_ast_unicode": "(∀([𝛗@Variable]:[Fm@Type])∀([x@Variable]:[Name@Type])∀([k@Variable]:[Class.Expression@Type])(Fm.FFI(([k@Class.Expression.Expression]:Class.Expression),‹[x@Name.Expression]›,[𝛗@Fm.Expression])→(𝘛,[]⊢‹(⟦[k@Class.Expression.Expression]⟧∈{[⦃[x@Variable.Expression]⦄@Class.Expression]|⟦[𝛗@Fm.Expression]⟧})⟷((⟦[k@Class.Expression.Expression]⟧ set)∧(let [x@Name.Expression] = ⟦[k@Class.Expression.Expression]⟧ in ⟦[𝛗@Fm.Expression]⟧))›)))",
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title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><a href=\"symbols/dot-delim.html\" class=\"sym-link\" title=\"Lambda binder body and sort projection separator (.)\">.</a><span class=\"ansi-light-green\"><a href=\"symbols/expression-sort.html\" class=\"sym-link\" title=\"Parameterized syntactic AST expression sort constructor (A.Expression)\">Expression</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-blue\"><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/dot-delim.html\" class=\"sym-link\" title=\"Lambda binder body and 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href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class expression 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class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "verification": "Postulated (Kernel Axiom)"
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    {
      "id": "sec-5",
      "title": "5. Incompleteness-Related Statements (5.01-5.02)",
      "range": "5.01-5.02",
      "description": "Formalized Löb's theorem and Gödel's second incompleteness theorem.",
      "entries": [
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      "index": 105,
      "name": "ThmLoeb",
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      "legacy_id": "item-5.01",
      "num": "5.01",
      "title": "Löb's theorem(WIP)",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
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href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-tau.html\" class=\"sym-link\" title=\"Recursively Enumerable Base Class Theory T (Constructor: HI IpTau)\">𝘛</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-tau.html\" class=\"sym-link\" title=\"Recursively Enumerable Base Class Theory T (Constructor: HI IpTau)\">𝘛</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛗\">𝛗</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛗\">𝛗</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-tau.html\" class=\"sym-link\" title=\"Recursively Enumerable Base Class Theory T (Constructor: HI IpTau)\">𝘛</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛗\">𝛗</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/typing-colon.html}{:} \\href{symbols/fm-sort.html}{\\mathsf{Fm}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-eau.html}{\\mathcal{E}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/quotation.html}{\\langle } \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/quotation.html}{\\langle } \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/quotation.html}{\\rangle } \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/quotation.html}{\\rangle } \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
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      "side_condition_unicode_html": "",
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      "comment": "Here 𝘌 ∈ {IΣ₁, PA, 𝘛, NBG, MK} is an unspecified elementary metatheory capable of proving the Hilbert-Bernays-Löb derivability conditions D1-D3 for the r.e. base theory 𝘛 (specifically D3, Σ₁-completeness, which requires Σ₁-induction and fails in Robinson's arithmetic Q; see Hájek & Pudlák 1993, Metamathematics of First-Order Arithmetic, Ch. I §3 & Ch. III; Bezboruah & Shepherdson 1976, JSL 41(2):503-512; Boolos 1993, The Logic of Provability, Ch. 2).",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 106,
      "name": "ThmSecondIncompleteness",
      "id": "ThmSecondIncompleteness",
      "legacy_id": "item-5.02",
      "num": "5.02",
      "title": "Gödel's second incompleteness theorem(WIP)",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
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      "prf_constructor": "TpHolds",
      "scheme_of_types_unicode": "(𝘌,[]⊢‹(¬(𝘛,[]⊢‹⊥›))→¬(𝘛,[]⊢‹¬(𝘛,[]⊢‹⊥›)›)›)",
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class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/neg-conn.html\" class=\"sym-link\" title=\"Logical negation connective (Constructor: HP (TpNeg ...))\">¬</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-tau.html\" class=\"sym-link\" title=\"Recursively Enumerable Base Class Theory T (Constructor: HI IpTau)\">𝘛</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a 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title=\"Logical negation connective (Constructor: HP (TpNeg ...))\">¬</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-tau.html\" class=\"sym-link\" title=\"Recursively Enumerable Base Class Theory T (Constructor: HI IpTau)\">𝘛</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><span class=\"ansi-bold-blue\"><a href=\"symbols/bot-const.html\" class=\"sym-link\" title=\"Falsum / logical falsity constant (Constructor: HP TpBot)\">⊥</a></span><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Formalized in the unspecified elementary metatheory 𝘌 ∈ {IΣ₁, PA, 𝘛, NBG, MK} (not Robinson's Q, which lacks Σ₁-induction needed for HBL condition D3; cf. Hájek & Pudlák 1993; Bezboruah & Shepherdson 1976) for the recursively enumerable object theory 𝘛. Note that Gödel's second incompleteness theorem applies to the r.e. theory 𝘛, whereas 𝘍 is non-r.e.",
      "verification": "Verified (Kernel)"
    }
      ]
    }
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      "name": "AxmPropWeakening",
      "id": "AxmPropWeakening",
      "legacy_id": "item-1.01",
      "num": "1.01",
      "title": "A01",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 2,
      "name": "AxmPropFrege",
      "id": "AxmPropFrege",
      "legacy_id": "item-1.02",
      "num": "1.02",
      "title": "A02",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛃\">𝛃</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" 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      "name": "AxmPropConjLeft",
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      "title": "A03",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 4,
      "name": "AxmPropConjRight",
      "id": "AxmPropConjRight",
      "legacy_id": "item-1.04",
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      "title": "A04",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 5,
      "name": "AxmPropConjIntro",
      "id": "AxmPropConjIntro",
      "legacy_id": "item-1.05",
      "num": "1.05",
      "title": "A05",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "side_condition_latex_linked": "",
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      "index": 6,
      "name": "AxmPropDisjLeft",
      "id": "AxmPropDisjLeft",
      "legacy_id": "item-1.06",
      "num": "1.06",
      "title": "A06",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 7,
      "name": "AxmPropDisjRight",
      "id": "AxmPropDisjRight",
      "legacy_id": "item-1.07",
      "num": "1.07",
      "title": "A07",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 8,
      "name": "AxmPropDisjElim",
      "id": "AxmPropDisjElim",
      "legacy_id": "item-1.08",
      "num": "1.08",
      "title": "A08",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/or-conn.html\" class=\"sym-link\" title=\"Logical disjunction connective (Constructor: HP (TpOr ...))\">∨</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-red\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛃\">𝛃</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a 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      "judgements_unicode_html": "<a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛃\">𝛃</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛄\">𝛄</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a>",
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      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Postulated (Kernel Axiom)"
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    {
      "index": 9,
      "name": "AxmPropNegElim",
      "id": "AxmPropNegElim",
      "legacy_id": "item-1.09",
      "num": "1.09",
      "title": "A09",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "((¬⟦𝛂⟧)→(⟦𝛂⟧→⊥))",
      "prf_constructor": "PA9_prf",
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      "judgements_unicode_html": "<a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a>",
      "judgements_latex_linked": "\\href{symbols/image-op.html}{[} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\alpha}} \\href{symbols/typing-colon.html}{:} \\href{symbols/fm-sort.html}{\\mathsf{Fm}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/image-op.html}{]} ",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Negation Elimination (Contradiction to Bottom): (¬⟦𝛂⟧) → (⟦𝛂⟧ → ⊥).",
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    },
    {
      "index": 10,
      "name": "AxmPropNegIntro",
      "id": "AxmPropNegIntro",
      "legacy_id": "item-1.10",
      "num": "1.10",
      "title": "A10",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "((⟦𝛂⟧→⟦𝛃⟧)→((⟦𝛂⟧→¬⟦𝛃⟧)→¬⟦𝛂⟧))",
      "prf_constructor": "PA10_prf",
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      "judgements_unicode": "[(𝛂:Fm), (𝛃:Fm)]",
      "judgements_unicode_html": "<a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛃\">𝛃</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a>",
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      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
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      "verification": "Postulated (Kernel Axiom)"
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    {
      "index": 11,
      "name": "AxmPropExFalso",
      "id": "AxmPropExFalso",
      "legacy_id": "item-1.11",
      "num": "1.11",
      "title": "A11",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "(⊥→⟦𝛂⟧)",
      "prf_constructor": "PA11_prf",
      "scheme_of_types_unicode": "(⊥→⟦[𝛂@Fm.Expression]⟧)",
      "kernel_ast_unicode": "(⊥→⟦[𝛂@Fm.Expression]⟧)",
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      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/bot-const.html}{\\bot} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\alpha}} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "[(𝛂:Fm)]",
      "judgements_unicode_html": "<a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Free formula variable 𝛂\">𝛂</a><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a>",
      "judgements_latex_linked": "\\href{symbols/image-op.html}{[} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\alpha}} \\href{symbols/typing-colon.html}{:} \\href{symbols/fm-sort.html}{\\mathsf{Fm}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/image-op.html}{]} ",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Postulated (Kernel Axiom)"
    },
    {
      "index": 12,
      "name": "AxmPropExcludedMiddle",
      "id": "AxmPropExcludedMiddle",
      "legacy_id": "item-1.12",
      "num": "1.12",
      "title": "A12",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "(⟦𝛂⟧∨¬⟦𝛂⟧)",
      "prf_constructor": "PA12_prf",
      "scheme_of_types_unicode": "(⟦[𝛂@Fm.Expression]⟧∨¬⟦[𝛂@Fm.Expression]⟧)",
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      "index": 18,
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      "index": 27,
      "name": "ThmEmptySetIsSubclass",
      "id": "ThmEmptySetIsSubclass",
      "legacy_id": "item-3.07",
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      "index": 29,
      "name": "AxmRightInSound",
      "id": "AxmRightInSound",
      "legacy_id": "item-3.09",
      "num": "3.09",
      "title": "“Membership respects equality from the right”",
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      "index": 31,
      "name": "AxmIntensionality",
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      "index": 32,
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      "index": 33,
      "name": "AxmExponent",
      "id": "AxmExponent",
      "legacy_id": "item-3.13",
      "num": "3.13",
      "title": "“Exponent”(powerclass of a set is also a set)",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 34,
      "name": "AxmRegularity",
      "id": "AxmRegularity",
      "legacy_id": "item-3.14",
      "num": "3.14",
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      "index": 35,
      "name": "DefTransitiveClass",
      "id": "DefTransitiveClass",
      "legacy_id": "item-3.15",
      "num": "3.15",
      "title": "Transitivity predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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      "prf_constructor": "PKernelDef_prf",
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      "side_condition_unicode_html": "",
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      "index": 36,
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      "id": "DefOrdinalClass",
      "legacy_id": "item-3.16",
      "num": "3.16",
      "title": "Ordinality predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "index": 37,
      "name": "DefClassOfOrdinals",
      "id": "DefClassOfOrdinals",
      "legacy_id": "item-3.17",
      "num": "3.17",
      "title": "Class of all ordinal sets(ordinal numbers)",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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      "legacy_id": "item-3.20",
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class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable b (binder)\">b</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-green\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable x\">x</a></span><span class=\"ansi-light-green\"><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable b\">b</a></span><span class=\"ansi-light-green\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable F\">F</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><span class=\"ansi-light-red\"><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension { x | ϗ } (AST constructor: CpKappa, binder)\">}</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{F} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/dom-op.html}{\\mathrm{Dom}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{F} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/eq-pred.html}{=} \\href{symbols/class-comprehension.html}{\\{} \\href{symbols/variables.html}{x} \\href{symbols/class-comprehension.html}{ \\mid } \\href{symbols/exists-binder.html}{\\exists} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{b} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{x} \\href{symbols/comma-delim.html}{,} \\href{symbols/variables.html}{b} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/in-pred.html}{\\in} \\href{symbols/variables.html}{F} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/class-comprehension.html}{\\}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Definitional (Kernel AST)"
    },
    {
      "index": 42,
      "name": "DefClassConditional",
      "id": "DefClassConditional",
      "legacy_id": "item-3.22",
      "num": "3.22",
      "title": "IF-THEN-ELSE-FI (Class conditional with NFV side condition and context turnstile)",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
      "formula_unicode": "(∀(Γ:Ctx)∀(v:Variable)∀(𝛙:Fm)∀(A:Class)∀(B:Class)((¬Fm.IFV(v,𝛙))→(𝘞,Γ⊢‹⟦IF ⟦𝛙⟧ THEN A ELSE B FI⟧={⦃v⦄|(⟦𝛙⟧∧(⦃v⦄∈A))∨((¬⟦𝛙⟧)∧(⦃v⦄∈B))}›)))",
      "prf_constructor": "CpIf",
      "scheme_of_types_unicode": "(∀([Γ@Variable]:[Ctx@Type])∀([v@Variable]:[Variable@Type])∀([𝛙@Variable]:[Fm@Type])∀([A@Variable]:[Class@Type])∀([B@Variable]:[Class@Type])((¬Fm.IFV([v@Variable.Expression],[𝛙@Fm.Expression]))→(𝘞,[Γ@Ctx.Expression]⊢‹⟦IF ⟦[𝛙@Fm.Expression]⟧ THEN [A@Class.Expression] ELSE [B@Class.Expression] FI⟧={[⦃[v@Variable.Expression]⦄@Class.Expression]|(⟦[𝛙@Fm.Expression]⟧∧([⦃[v@Variable.Expression]⦄@Class.Expression]∈[A@Class.Expression]))∨((¬⟦[𝛙@Fm.Expression]⟧)∧([⦃[v@Variable.Expression]⦄@Class.Expression]∈[B@Class.Expression]))}›)))",
      "kernel_ast_unicode": "(∀([Γ@Variable]:[Ctx@Type])∀([v@Variable]:[Variable@Type])∀([𝛙@Variable]:[Fm@Type])∀([A@Variable]:[Class@Type])∀([B@Variable]:[Class@Type])((¬Fm.IFV([v@Variable.Expression],[𝛙@Fm.Expression]))→(𝘞,[Γ@Ctx.Expression]⊢‹⟦IF ⟦[𝛙@Fm.Expression]⟧ THEN [A@Class.Expression] ELSE [B@Class.Expression] FI⟧={[⦃[v@Variable.Expression]⦄@Class.Expression]|(⟦[𝛙@Fm.Expression]⟧∧([⦃[v@Variable.Expression]⦄@Class.Expression]∈[A@Class.Expression]))∨((¬⟦[𝛙@Fm.Expression]⟧)∧([⦃[v@Variable.Expression]⦄@Class.Expression]∈[B@Class.Expression]))}›)))",
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title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable A (binder)\">A</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B (binder)\">B</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/neg-conn.html\" class=\"sym-link\" title=\"Logical negation connective (Constructor: HP (TpNeg ...))\">¬</a><span class=\"ansi-bold-blue\"><a href=\"symbols/fm-sort.html\" class=\"sym-link\" title=\"Object Formula syntactic sort (Fm = Bool.Expression)\">Fm</a><a href=\"symbols/dot-delim.html\" class=\"sym-link\" title=\"Lambda binder body and sort projection separator (.)\">.</a><a href=\"symbols/ifv-pred.html\" class=\"sym-link\" title=\"Is Free Variable binary predicate symbol (Fm.IFV, Ctx.IFV, ClassTerm.IFV; Constructor: HP (TpIFV ...))\">IFV</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable v\">v</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛙\">𝛙</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-wau.html\" class=\"sym-link\" title=\"Free theory variable 𝘞\">𝘞</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound context variable Γ\">Γ</a></span><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP 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title=\"Class conditional IF-THEN-ELSE-FI functional symbol (Constructor: EIf / CpIf)\">THEN</a> <span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable A\">A</a></span> <a href=\"symbols/class-if-then-else.html\" class=\"sym-link\" title=\"Class conditional IF-THEN-ELSE-FI functional symbol (Constructor: EIf / CpIf)\">ELSE</a> <span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B\">B</a></span> <a href=\"symbols/class-if-then-else.html\" class=\"sym-link\" title=\"Class conditional IF-THEN-ELSE-FI functional symbol (Constructor: EIf / CpIf)\">FI</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><span class=\"ansi-light-red\"><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension { x | ϗ } (AST constructor: CpKappa, binder)\">{</a></span><span class=\"ansi-white\"><a href=\"symbols/binder-brackets.html\" class=\"sym-link\" title=\"Left syntactic binder-variable bracket {| x |} (Constructor: IVariable_t)\">⦃</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable v\">v</a></span><span class=\"ansi-white\"><a href=\"symbols/binder-brackets.html\" class=\"sym-link\" title=\"Right syntactic binder-variable bracket {| x |} (Constructor: IVariable_t)\">⦄</a></span><span class=\"ansi-light-red\"><a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension separator | in { x | ϗ } (AST constructor: CpKappa, binder)\">|</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛙\">𝛙</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-white\"><a href=\"symbols/binder-brackets.html\" class=\"sym-link\" title=\"Left syntactic binder-variable bracket {| x |} (Constructor: IVariable_t)\">⦃</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable v\">v</a></span><span class=\"ansi-white\"><a href=\"symbols/binder-brackets.html\" class=\"sym-link\" title=\"Right syntactic binder-variable bracket {| x |} (Constructor: IVariable_t)\">⦄</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable A\">A</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/or-conn.html\" class=\"sym-link\" title=\"Logical disjunction connective (Constructor: HP (TpOr ...))\">∨</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/neg-conn.html\" class=\"sym-link\" title=\"Logical negation connective (Constructor: HP (TpNeg ...))\">¬</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟦</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛙\">𝛙</a></span><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Right semantic evaluation / antiquotation bracket (Constructor: EAnti / EEval)\">⟧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-white\"><a href=\"symbols/binder-brackets.html\" class=\"sym-link\" title=\"Left syntactic binder-variable bracket {| x |} (Constructor: IVariable_t)\">⦃</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable v\">v</a></span><span class=\"ansi-white\"><a href=\"symbols/binder-brackets.html\" class=\"sym-link\" title=\"Right syntactic binder-variable bracket {| x |} (Constructor: IVariable_t)\">⦄</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a 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      "index": 48,
      "name": "DefIsSetValued",
      "id": "DefIsSetValued",
      "legacy_id": "item-3.28",
      "num": "3.28",
      "title": "“Is set-valued” predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable x\">x</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/dom-op.html\" class=\"sym-link\" title=\"Domain unary functional symbol 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class=\"sym-link\" title=\"Bound class variable F\">F</a></span><span class=\"ansi-light-magenta\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable x\">x</a></span><span class=\"ansi-light-magenta\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span> <a href=\"symbols/set-pred.html\" class=\"sym-link\" title=\"Is a set unary predicate symbol (Constructor: HP (TpSet ...))\">set</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
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      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
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      "index": 49,
      "name": "DefHasConcentratedValues",
      "id": "DefHasConcentratedValues",
      "legacy_id": "item-3.29",
      "num": "3.29",
      "title": "“Has concentrated set values” predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "index": 50,
      "name": "DefFlatValues",
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      "num": "3.30",
      "title": "“Has flat proper values” predicate",
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      "status": "Definitional (Kernel AST)",
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href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-green\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / 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title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable b\">b</a></span> <a href=\"symbols/singleton-pred.html\" class=\"sym-link\" title=\"Has exactly one element unary predicate symbol (Constructor: HP (TpSingleton ...))\">singleton</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "index": 51,
      "name": "DefFamily",
      "id": "DefFamily",
      "legacy_id": "item-3.31",
      "num": "3.31",
      "title": "“Is a family” predicate",
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      "status": "Definitional (Kernel AST)",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
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      "index": 52,
      "name": "DefClassFunction",
      "id": "DefClassFunction",
      "legacy_id": "item-3.32",
      "num": "3.32",
      "title": "“Is a class-function” predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
      "formula_unicode": "(∀(F:Class)((F class-function)⟷((F family)∧(F set-valued))))",
      "prf_constructor": "TpClassFunc",
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      "kernel_ast_unicode": "(∀([F@Variable]:[Class@Type])(([F@Class.Expression] class-function)⟷(([F@Class.Expression] family)∧([F@Class.Expression] set-valued))))",
      "formula_unicode_html": "<a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable F (binder)\">F</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable F\">F</a></span> <a href=\"symbols/class-func-pred.html\" class=\"sym-link\" title=\"Is a class-function unary predicate symbol (Constructor: HP (TpClassFunc ...))\">class-function</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/iff-conn.html\" class=\"sym-link\" title=\"Logical equivalence connective (Constructor: HP (TpIff ...))\">⟷</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable F\">F</a></span> <a href=\"symbols/family-pred.html\" class=\"sym-link\" title=\"Is a family unary predicate symbol (Constructor: HP (TpFamily ...))\">family</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable F\">F</a></span> <a href=\"symbols/set-valued-pred.html\" class=\"sym-link\" title=\"Is set-valued unary predicate symbol (Constructor: HP (TpSetValued ...))\">set-valued</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
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      "side_condition_latex_linked": "",
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    {
      "index": 53,
      "name": "DefInjectiveFunction",
      "id": "DefInjectiveFunction",
      "legacy_id": "item-3.33",
      "num": "3.33",
      "title": "“Is injective” predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
      "formula_unicode": "(∀(F:Class)((F injective)⟷(((F family)∧(F set-valued))∧∀(x:Class)∀(y:Class)((((x∈Dom(F))∧(y∈Dom(F)))∧(F(x)=F(y)))→(x=y)))))",
      "prf_constructor": "TpInjective",
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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable F\">F</a></span><span class=\"ansi-light-magenta\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable y\">y</a></span><span class=\"ansi-light-magenta\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" 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grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable a\">a</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/dom-op.html\" class=\"sym-link\" title=\"Domain unary functional symbol (Constructor: HC (CpDom ...))\">Dom</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B\">B</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable b\">b</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B\">B</a></span><span class=\"ansi-light-magenta\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span 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class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "comment": "Explicit typed term interpretation / valuation (μ_A : A.Expression → Class.Expression) of the dependent sum Σ(x:Dom(F)) over a family F in the fixed background model.",
      "verification": "Definitional (Kernel AST)"
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    {
      "index": 59,
      "name": "DefPiPredicateModel",
      "id": "DefPiPredicateModel",
      "legacy_id": "item-3.39",
      "num": "3.39",
      "title": "“Dependent Product (Pi-Type)” predicate and section model",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
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      "index": 60,
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      "index": 61,
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      "index": 65,
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      "legacy_id": "item-3.45",
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href=\"symbols/iff-conn.html\" class=\"sym-link\" title=\"Logical equivalence connective (Constructor: HP (TpIff ...))\">⟷</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B (binder)\">B</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B\">B</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/power-op.html\" class=\"sym-link\" title=\"Powerclass / Exponent unary functional symbol (Constructor: HC (CpPower ...))\">𝒫</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B\">B</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "index": 66,
      "name": "DefPairfulClass",
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      "legacy_id": "item-3.46",
      "num": "3.46",
      "title": "“Is pairful” predicate",
      "kind": "Definition",
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      "index": 70,
      "name": "DefNaturalNumberModels",
      "id": "DefNaturalNumberModels",
      "legacy_id": "item-3.50",
      "num": "3.50",
      "title": "Set-theoretic models of natural numbers(TBI)",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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    {
      "index": 71,
      "name": "DefClassLists",
      "id": "DefClassLists",
      "legacy_id": "item-3.51",
      "num": "3.51",
      "title": "General definition of lists of classes which applicable not only for sets, but for proper classes too(TBI)",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a> <a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a> <a href=\"symbols/emptyset-const.html\" class=\"sym-link\" title=\"Empty set / empty class constant (Constructor: HC CpEmptySet)\">∅</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a> <a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Left relational image f[A] (CpImage) / Dunaev class-list bracket\">[</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free class variable A\">A</a></span><a href=\"symbols/image-op.html\" class=\"sym-link\" title=\"Right relational image f[A] (CpImage) / Dunaev class-list bracket\">]</a> <a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a> <a href=\"symbols/mark-op.html\" class=\"sym-link\" title=\"Dunaev class-list mark functional symbol (mark)\">mark</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free class variable A\">A</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/numeral-1.html\" class=\"sym-link\" title=\"von Neumann natural number 1 = {0}\">1</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a> <a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / 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parenthesis\">(</a><a href=\"symbols/mark-op.html\" class=\"sym-link\" title=\"Dunaev class-list mark functional symbol (mark)\">mark</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free class variable A\">A</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/numeral-1.html\" class=\"sym-link\" title=\"von Neumann natural number 1 = {0}\">1</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a> <a href=\"symbols/binunion-op.html\" class=\"sym-link\" title=\"Binary union functional symbol (Constructor: HC (CpBinUnion ...))\">∪</a> <a href=\"symbols/mark-op.html\" class=\"sym-link\" title=\"Dunaev class-list mark functional symbol (mark)\">mark</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free class variable B\">B</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/numeral-2.html\" class=\"sym-link\" title=\"von Neumann natural number 2 = {0,1}\">2</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered 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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a> <a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/pr1-op.html\" class=\"sym-link\" title=\"First projection functional symbol of Dunaev pair (pr1)\">pr1</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free class variable L\">L</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a> <a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a> <a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension { x | ϗ } (AST constructor: CpKappa, binder)\">{</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable a (binder)\">a</a></span> <a href=\"symbols/class-comprehension.html\" class=\"sym-link\" title=\"Class comprehension separator | in { x | ϗ } (AST constructor: CpKappa, binder)\">|</a> <a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/upair-op.html\" class=\"sym-link\" title=\"Unordered pair {a, b} (AST constructor: CpUPair, 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href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{x} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{x} \\href{symbols/in-pred.html}{\\in} \\href{symbols/omega-const.html}{\\omega} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/iff-conn.html}{\\longleftrightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{Q} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/ind-pred.html}{\\mathrm{Ind}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{Q} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{x} \\href{symbols/in-pred.html}{\\in} \\href{symbols/variables.html}{Q} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "By definition, ω is not only the smallest successor-inductive ordinal, but also the ⊆-minimal (least) element among all successor-inductive (Peano-inductive) classes in the subclass (⊆) order (Ind(ω) ∧ ∀(Q:Class), Ind(Q) → (ω ⊆ Q)), directly expressing simple countable mathematical (Peano) induction up to ω. | Special Def.: ω = ⋂{q | Ind(q)}; General Def.: ∀(x:Class), (x ∈ ω) ⟷ (∀(Q:Class), Ind(Q) → (x ∈ Q)).",
      "verification": "Definitional (Kernel AST)"
    },
    {
      "index": 77,
      "name": "AxmOmegaIsSet",
      "id": "AxmOmegaIsSet",
      "legacy_id": "item-3.57",
      "num": "3.57",
      "title": "“Infinity”(ω is a set)",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "(ω set)",
      "prf_constructor": "POmegaSetAx_prf",
      "scheme_of_types_unicode": "(ω set)",
      "kernel_ast_unicode": "(ω set)",
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      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/omega-const.html}{\\omega}  \\href{symbols/set-pred.html}{\\mathsf{set}} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Asserts that the ⊆-minimal successor-inductive (Peano-inductive) class ω is a set.",
      "verification": "Postulated (Kernel Axiom)"
    },
    {
      "index": 78,
      "name": "ThmUniversalClassInductive",
      "id": "ThmUniversalClassInductive",
      "legacy_id": "item-3.58",
      "num": "3.58",
      "title": "Class of all sets is successor-inductive (Peano-inductive)",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Ind(V)",
      "prf_constructor": "TpInd",
      "scheme_of_types_unicode": "Ind(V)",
      "kernel_ast_unicode": "Ind(V)",
      "formula_unicode_html": "<span class=\"ansi-bold-blue\"><a href=\"symbols/ind-pred.html\" class=\"sym-link\" title=\"Is successor-inductive (Peano-inductive) unary predicate symbol (Constructor: HP (TpInd ...))\">Ind</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/universal-class-const.html\" class=\"sym-link\" title=\"Universal class V constant (Constructor: HC CpSets)\">V</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/ind-pred.html}{\\mathrm{Ind}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/universal-class-const.html}{\\mathbf{V}} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 79,
      "name": "ThmOmegaInductive",
      "id": "ThmOmegaInductive",
      "legacy_id": "item-3.59",
      "num": "3.59",
      "title": "Intersection of all successor-inductive (Peano-inductive) classes is a successor-inductive class itself (ω is the ⊆-least successor-inductive class)",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Ind(ω)",
      "prf_constructor": "TpInd",
      "scheme_of_types_unicode": "Ind(ω)",
      "kernel_ast_unicode": "Ind(ω)",
      "formula_unicode_html": "<span class=\"ansi-bold-blue\"><a href=\"symbols/ind-pred.html\" class=\"sym-link\" title=\"Is successor-inductive (Peano-inductive) unary predicate symbol (Constructor: HP (TpInd ...))\">Ind</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/omega-const.html\" class=\"sym-link\" title=\"Smallest successor-inductive ordinal and subseteq-minimal successor-inductive class (Constructor: HC CpOmega)\">ω</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/ind-pred.html}{\\mathrm{Ind}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/omega-const.html}{\\omega} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Establishes that ω itself satisfies Ind(ω), making ω the ⊆-least (minimal) element among all successor-inductive classes in the ⊆ order.",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 80,
      "name": "ThmOrdinalsInductive",
      "id": "ThmOrdinalsInductive",
      "legacy_id": "item-3.60",
      "num": "3.60",
      "title": "Class of all ordinal numbers is successor-inductive (Peano-inductive)",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Ind(On)",
      "prf_constructor": "TpInd",
      "scheme_of_types_unicode": "Ind(On)",
      "kernel_ast_unicode": "Ind(On)",
      "formula_unicode_html": "<span class=\"ansi-bold-blue\"><a href=\"symbols/ind-pred.html\" class=\"sym-link\" title=\"Is successor-inductive (Peano-inductive) unary predicate symbol (Constructor: HP (TpInd ...))\">Ind</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/on-const.html\" class=\"sym-link\" title=\"Class of all ordinal sets On constant (Constructor: HC CpOn)\">On</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/ind-pred.html}{\\mathrm{Ind}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/on-const.html}{\\mathbf{On}} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 81,
      "name": "ThmOmegaTransitive",
      "id": "ThmOmegaTransitive",
      "legacy_id": "item-3.61",
      "num": "3.61",
      "title": "ω is transitive",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Tr(ω)",
      "prf_constructor": "TpTrans",
      "scheme_of_types_unicode": "Tr(ω)",
      "kernel_ast_unicode": "Tr(ω)",
      "formula_unicode_html": "<span class=\"ansi-bold-blue\"><a href=\"symbols/trans-pred.html\" class=\"sym-link\" title=\"Transitivity unary predicate symbol (Constructor: HP (TpTrans ...))\">Tr</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/omega-const.html\" class=\"sym-link\" title=\"Smallest successor-inductive ordinal and subseteq-minimal successor-inductive class (Constructor: HC CpOmega)\">ω</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/trans-pred.html}{\\mathrm{Tr}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/omega-const.html}{\\omega} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 82,
      "name": "ThmOmegaOrdinal",
      "id": "ThmOmegaOrdinal",
      "legacy_id": "item-3.62",
      "num": "3.62",
      "title": "ω is ordinal",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Ord(ω)",
      "prf_constructor": "TpOrd",
      "scheme_of_types_unicode": "Ord(ω)",
      "kernel_ast_unicode": "Ord(ω)",
      "formula_unicode_html": "<span class=\"ansi-bold-blue\"><a href=\"symbols/ord-pred.html\" class=\"sym-link\" title=\"Ordinality unary predicate symbol (Constructor: HP (TpOrd ...))\">Ord</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/omega-const.html\" class=\"sym-link\" title=\"Smallest successor-inductive ordinal and subseteq-minimal successor-inductive class (Constructor: HC CpOmega)\">ω</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/ord-pred.html}{\\mathrm{Ord}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/omega-const.html}{\\omega} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 83,
      "name": "ThmOmegaInOrdinals",
      "id": "ThmOmegaInOrdinals",
      "legacy_id": "item-3.63",
      "num": "3.63",
      "title": "ω is in the class of all ordinal numbers(=sets)",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "(ω∈On)",
      "prf_constructor": "CpOn",
      "scheme_of_types_unicode": "(ω∈On)",
      "kernel_ast_unicode": "(ω∈On)",
      "formula_unicode_html": "<a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/omega-const.html\" class=\"sym-link\" title=\"Smallest successor-inductive ordinal and subseteq-minimal successor-inductive class (Constructor: HC CpOmega)\">ω</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/on-const.html\" class=\"sym-link\" title=\"Class of all ordinal sets On constant (Constructor: HC CpOn)\">On</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/omega-const.html}{\\omega} \\href{symbols/in-pred.html}{\\in} \\href{symbols/on-const.html}{\\mathbf{On}} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
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    {
      "index": 84,
      "name": "DefIsCardinal",
      "id": "DefIsCardinal",
      "legacy_id": "item-3.64",
      "num": "3.64",
      "title": "“Is cardinal” predicate",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "(∀(C:Class)(Card(C)⟷(∃(K:Class)(|K|=C))))",
      "prf_constructor": "TpCard",
      "scheme_of_types_unicode": "(∀([C@Variable]:[Class@Type])(Card([C@Class.Expression])⟷(∃([K@Variable]:[Class@Type])(|[K@Class.Expression]|=[C@Class.Expression]))))",
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      "formula_unicode_html": "<a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable C (binder)\">C</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-blue\"><a href=\"symbols/card-pred.html\" class=\"sym-link\" title=\"Is a cardinal unary predicate symbol (Constructor: HP (TpCard ...))\">Card</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable C\">C</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/iff-conn.html\" class=\"sym-link\" title=\"Logical equivalence connective (Constructor: HP (TpIff ...))\">⟷</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/exists-binder.html\" class=\"sym-link\" title=\"Existential quantifier binder (Constructor: HP (TpExis ...))\">∃</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable K (binder)\">K</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/card-op.html\" class=\"sym-link\" title=\"Cardinality / equinumerosity |C| (AST constructor: CpCard, functional symbol)\">|</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable K\">K</a></span><a href=\"symbols/card-op.html\" class=\"sym-link\" title=\"Cardinality / equinumerosity |C| (AST constructor: CpCard, functional symbol)\">|</a><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable C\">C</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{C} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/card-pred.html}{\\mathrm{Card}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{C} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/iff-conn.html}{\\longleftrightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/exists-binder.html}{\\exists} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{K} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/card-op.html}{ \\mid } \\href{symbols/variables.html}{K} \\href{symbols/card-op.html}{ \\mid } \\href{symbols/eq-pred.html}{=} \\href{symbols/variables.html}{C} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 85,
      "name": "ThmOmegaCardinal",
      "id": "ThmOmegaCardinal",
      "legacy_id": "item-3.65",
      "num": "3.65",
      "title": "ω is cardinal",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Card(ω)",
      "prf_constructor": "TpCard",
      "scheme_of_types_unicode": "Card(ω)",
      "kernel_ast_unicode": "Card(ω)",
      "formula_unicode_html": "<span class=\"ansi-bold-blue\"><a href=\"symbols/card-pred.html\" class=\"sym-link\" title=\"Is a cardinal unary predicate symbol (Constructor: HP (TpCard ...))\">Card</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/omega-const.html\" class=\"sym-link\" title=\"Smallest successor-inductive ordinal and subseteq-minimal successor-inductive class (Constructor: HC CpOmega)\">ω</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/card-pred.html}{\\mathrm{Card}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/omega-const.html}{\\omega} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 86,
      "name": "ThmOrdinalsCardinal",
      "id": "ThmOrdinalsCardinal",
      "legacy_id": "item-3.66",
      "num": "3.66",
      "title": "On is a cardinal class",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "Card(On)",
      "prf_constructor": "TpCard",
      "scheme_of_types_unicode": "Card(On)",
      "kernel_ast_unicode": "Card(On)",
      "formula_unicode_html": "<span class=\"ansi-bold-blue\"><a href=\"symbols/card-pred.html\" class=\"sym-link\" title=\"Is a cardinal unary predicate symbol (Constructor: HP (TpCard ...))\">Card</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/on-const.html\" class=\"sym-link\" title=\"Class of all ordinal sets On constant (Constructor: HC CpOn)\">On</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 87,
      "name": "ThmUniversalClassNotCardinal",
      "id": "ThmUniversalClassNotCardinal",
      "legacy_id": "item-3.67",
      "num": "3.67",
      "title": "Class of all sets is not a cardinal class",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "(¬Card(V))",
      "prf_constructor": "TpCard",
      "scheme_of_types_unicode": "(¬Card(V))",
      "kernel_ast_unicode": "(¬Card(V))",
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      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/neg-conn.html}{\\neg} \\href{symbols/card-pred.html}{\\mathrm{Card}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/universal-class-const.html}{\\mathbf{V}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 88,
      "name": "DefGrothendieckUniverse",
      "id": "DefGrothendieckUniverse",
      "legacy_id": "item-3.68",
      "num": "3.68",
      "title": "“Is a universe” predicate",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
      "formula_unicode": "(∀(U:Class)((U universe)⟷(((ω∈U)∧((Tr(U)∧Hereditary(U))∧(Powerful(U)∧Tight(U))))∧Pairful(U))))",
      "prf_constructor": "PKernelDef_prf",
      "scheme_of_types_unicode": "(∀([U@Variable]:[Class@Type])(([U@Class.Expression] universe)⟷(((ω∈[U@Class.Expression])∧((Tr([U@Class.Expression])∧Hereditary([U@Class.Expression]))∧(Powerful([U@Class.Expression])∧Tight([U@Class.Expression]))))∧Pairful([U@Class.Expression]))))",
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href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-blue\"><a href=\"symbols/trans-pred.html\" class=\"sym-link\" title=\"Transitivity unary predicate symbol (Constructor: HP (TpTrans ...))\">Tr</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><span class=\"ansi-bold-blue\"><a href=\"symbols/hereditary-pred.html\" class=\"sym-link\" title=\"Is hereditary unary predicate symbol (Constructor: HP (TpHereditary ...))\">Hereditary</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-blue\"><a href=\"symbols/powerful-pred.html\" class=\"sym-link\" title=\"Is powerful unary predicate symbol (Constructor: HP (TpPowerful ...))\">Powerful</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><span class=\"ansi-bold-blue\"><a href=\"symbols/tight-pred.html\" class=\"sym-link\" title=\"Is tight unary predicate symbol (Constructor: HP (TpTight ...))\">Tight</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><span class=\"ansi-bold-blue\"><a href=\"symbols/pairful-pred.html\" class=\"sym-link\" title=\"Is pairful unary predicate symbol (Constructor: HP (TpPairful ...))\">Pairful</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable U\">U</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{U} \\href{symbols/typing-colon.html}{:} \\href{symbols/class-type.html}{\\mathsf{Class}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{U}  \\href{symbols/universe-pred.html}{\\mathsf{universe}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/iff-conn.html}{\\longleftrightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/omega-const.html}{\\omega} \\href{symbols/in-pred.html}{\\in} \\href{symbols/variables.html}{U} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/and-conn.html}{\\land} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/trans-pred.html}{\\mathrm{Tr}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{U} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/and-conn.html}{\\land} \\href{symbols/hereditary-pred.html}{\\mathsf{Hereditary}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{U} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/and-conn.html}{\\land} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/powerful-pred.html}{\\mathsf{Powerful}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{U} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/and-conn.html}{\\land} \\href{symbols/tight-pred.html}{\\mathsf{Tight}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{U} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/and-conn.html}{\\land} \\href{symbols/pairful-pred.html}{\\mathsf{Pairful}} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/variables.html}{U} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "",
      "verification": "Definitional (Kernel AST)"
    },
    {
      "index": 89,
      "name": "ThmUniversalClassIsUniverse",
      "id": "ThmUniversalClassIsUniverse",
      "legacy_id": "item-3.69",
      "num": "3.69",
      "title": "The class of all sets is a universe",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "(V universe)",
      "prf_constructor": "PVUniverseAx_prf",
      "scheme_of_types_unicode": "(V universe)",
      "kernel_ast_unicode": "(V universe)",
      "formula_unicode_html": "<a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-bold-yellow\"><a href=\"symbols/universal-class-const.html\" class=\"sym-link\" title=\"Universal class V constant (Constructor: HC CpSets)\">V</a></span> <a href=\"symbols/universe-pred.html\" class=\"sym-link\" title=\"Is a Grothendieck/Tarski universe unary predicate symbol (Constructor: HP (TpUniverse ...))\">universe</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/universal-class-const.html}{\\mathbf{V}}  \\href{symbols/universe-pred.html}{\\mathsf{universe}} \\href{symbols/delimiters-brackets.html}{)} ",
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      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Formal kernel proof (v_universe_thm_proof : unit -> Thm, verified by Thm.checkPipeline and Thm.slow_check) deriving (𝘍, [] ⊢ (V universe)) from 3.68 #DefGrothendieckUniverse (PKernelDef_prf) and the 6 component lemmas: (1) (ω ∈ V) via 3.57 #AxmOmegaIsSet (POmegaSetAx_prf), 3.05 #DefUniversalClass (PKernelDef_prf), 3.08 #AxmLeftInSound (PLeftInSoundAx_prf), and 3.01 #RulClassComprehension (PComprAx_prf); (2) (V transitive) via 3.15 #DefTransitiveClass (PKernelDef_prf), 3.06 #DefSubclass (PKernelDef_prf), and 3.02 #DefIsSet (PSetDef_prf); (3) (V Hereditary) via 3.44 #DefHereditaryClass (PKernelDef_prf) and 3.12 #AxmHeredity (PHeredityAx_prf); (4) (V Powerful) via 3.45 #DefPowerfulClass (PKernelDef_prf) and 3.13 #AxmExponent (PPowSetAx_prf); (5) (V Tight) via 3.47 #DefTightClass (PKernelDef_prf), 3.43 #AxmSizeLimitation (PSizeLimAx_prf), and 3.41/3.42 #DefCardinality/#DefCardLess (PCardLessOrderAx_prf); (6) (V Pairful) via 3.46 #DefPairfulClass (PKernelDef_prf) and 3.49 #DefUnorderedPair sethood (PPairSetAx_prf). Backup axiom constructor PVUniverseAx_prf (v_universe_axm_backup_entry) is also retained in the kernel just in case.",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 90,
      "name": "AxmTarskiUniverses",
      "id": "AxmTarskiUniverses",
      "legacy_id": "item-3.70",
      "num": "3.70",
      "title": "“Tarski's axiom”",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
      "formula_unicode": "(∀(x:Class)((x set)→∃(u:Class)(((u universe)∧(x∈u))∧(u set))))",
      "prf_constructor": "TpUniverse",
      "scheme_of_types_unicode": "(∀([x@Variable]:[Class@Type])(([x@Class.Expression] set)→∃([u@Variable]:[Class@Type])((([u@Class.Expression] universe)∧([x@Class.Expression]∈[u@Class.Expression]))∧([u@Class.Expression] set))))",
      "kernel_ast_unicode": "(∀([x@Variable]:[Class@Type])(([x@Class.Expression] set)→∃([u@Variable]:[Class@Type])((([u@Class.Expression] universe)∧([x@Class.Expression]∈[u@Class.Expression]))∧([u@Class.Expression] set))))",
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title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/exists-binder.html\" class=\"sym-link\" title=\"Existential quantifier binder (Constructor: HP (TpExis ...))\">∃</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable u (binder)\">u</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable u\">u</a></span> <a href=\"symbols/universe-pred.html\" class=\"sym-link\" title=\"Is a Grothendieck/Tarski universe unary predicate symbol (Constructor: HP (TpUniverse ...))\">universe</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable x\">x</a></span><a href=\"symbols/in-pred.html\" class=\"sym-link\" title=\"Class membership binary predicate symbol (Constructor: HP (TpIn ...))\">∈</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable u\">u</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/and-conn.html\" class=\"sym-link\" title=\"Logical conjunction connective (Constructor: HP (TpAnd ...))\">∧</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable u\">u</a></span> <a href=\"symbols/set-pred.html\" class=\"sym-link\" title=\"Is a set unary predicate symbol (Constructor: HP (TpSet ...))\">set</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "num": "3.71",
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class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "index": 92,
      "name": "AxmSoundnessOfBaseTheory",
      "id": "AxmSoundnessOfBaseTheory",
      "legacy_id": "item-3.72",
      "num": "3.72",
      "title": "Soundness of the theory 𝘛",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 93,
      "name": "AxmSoundnessOfFullTheory",
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      "legacy_id": "item-3.73",
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      "index": 95,
      "name": "AxmBaseSubtheoryOfFull",
      "id": "AxmBaseSubtheoryOfFull",
      "legacy_id": "item-3.75",
      "num": "3.75",
      "title": "If a formula is derivable in 𝘛, then it is derivable in 𝘍",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "index": 96,
      "name": "DefDunaevOrderedPair",
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      "legacy_id": "item-3.76",
      "num": "3.76",
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      "index": 97,
      "name": "ThmDunaevFirstProjection",
      "id": "ThmDunaevFirstProjection",
      "legacy_id": "item-3.77",
      "num": "3.77",
      "title": "First projection of Dunaev's tagged ordered pair",
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      "status": "Verified (Kernel)",
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      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
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    {
      "index": 98,
      "name": "ThmDunaevSecondProjection",
      "id": "ThmDunaevSecondProjection",
      "legacy_id": "item-3.78",
      "num": "3.78",
      "title": "Second projection of Dunaev's tagged ordered pair",
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      "status": "Verified (Kernel)",
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href=\"symbols/singleton-op.html\" class=\"sym-link\" title=\"Singleton set {a} (AST constructor: CpSingleton, functional symbol)\">}</a></span><span class=\"ansi-light-green\"><a href=\"symbols/singleton-op.html\" class=\"sym-link\" title=\"Singleton set {a} (AST constructor: CpSingleton, functional symbol)\">}</a></span><span class=\"ansi-light-green\"><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a></span><span class=\"ansi-light-green\"><a href=\"symbols/upair-op.html\" class=\"sym-link\" title=\"Unordered pair {a, b} (AST constructor: CpUPair, functional symbol)\">{</a></span><span class=\"ansi-bold-yellow\"><a href=\"symbols/emptyset-const.html\" class=\"sym-link\" title=\"Empty set / empty class constant (Constructor: HC CpEmptySet)\">∅</a></span><span class=\"ansi-light-green\"><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma 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constructor: CpKappa, binder)\">}</a></span><a href=\"symbols/eq-pred.html\" class=\"sym-link\" title=\"Extensional equality binary predicate symbol (Constructor: HP (TpEq ...))\">=</a><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable B\">B</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
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      "judgements_latex_linked": "",
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      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Recovers B from (A,B) via tag 2 without requiring A or B to be sets.",
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    {
      "index": 99,
      "name": "DefMetaInstantiation",
      "id": "DefMetaInstantiation",
      "legacy_id": "item-4.01",
      "num": "4.01",
      "title": "Instantiation(meta,WIP)",
      "kind": "Axiom",
      "status": "Postulated (Kernel Axiom)",
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      "prf_constructor": "ELet",
      "scheme_of_types_unicode": "(∀([A@Variable]:[Type@Type])∀([G@Variable]:[Ctx.Expression@Type])∀([𝛗@Variable]:[Fm@Type])∀([t@Variable]:[eval([A@Type.Expression.Expression]).Expression@Type])∀([v@Variable]:[Variable@Type])(𝘞,⟦[G@Ctx.Expression.Expression]⟧⊢‹(∀([⦃[v@Variable.Expression]⦄@Variable]:[⟦[A@Type.Expression.Expression]⟧@Type])⟦[𝛗@Fm.Expression]⟧)→(let ⟦[v@Name.Expression]⟧ = ⟦[t@Class.Expression.Expression]⟧ in ⟦[𝛗@Fm.Expression]⟧)›))",
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expression variable t (binder)\">t</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><a href=\"symbols/eval.html\" class=\"sym-link\" title=\"Metatheoretic formula valuation / truth evaluation (eval)\">eval</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-red\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Free type variable A\">A</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/dot-delim.html\" class=\"sym-link\" title=\"Lambda binder body and sort projection separator (.)\">.</a><span class=\"ansi-light-green\"><a href=\"symbols/expression-sort.html\" class=\"sym-link\" title=\"Parameterized syntactic AST expression sort constructor (A.Expression)\">Expression</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound variable-name variable v (binder)\">v</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span 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      "name": "DefMetaGeneralisation",
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      "num": "4.02",
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      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "(A theory 𝘞 is assumed here to contain only formulas without free variables, so the variable x has no free occurences in any formula of a set of formulas [|Dec(Theory.Expression)(W)|]. Here Enc(v) (μ_Variable(v) : Variable.Expression → Class.Expression) denotes the explicit typed term interpretation / valuation homomorphism mapping a typed term of sort A into a term of sort Class while respecting the fixed model and variable valuation.)",
      "verification": "Postulated (Kernel Axiom)"
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      "index": 101,
      "name": "DefMetaConecessitation",
      "id": "DefMetaConecessitation",
      "legacy_id": "item-4.03",
      "num": "4.03",
      "title": "CoNecessitation(meta,WIP)",
      "kind": "Inference Rule",
      "status": "Primitive / Derived Rule (Kernel)",
      "formula_unicode": "(∀(Γ:Ctx)∀(𝛗:Fm)((𝘍,[]⊢‹𝘍,Γ⊢𝛗›)→(𝘍,Γ⊢𝛗)))",
      "prf_constructor": "PCredo_prf",
      "scheme_of_types_unicode": "(∀([Γ@Variable]:[Ctx@Type])∀([𝛗@Variable]:[Fm@Type])((𝘍,[]⊢‹𝘍,[Γ@Ctx.Expression]⊢[𝛗@Fm.Expression]›)→(𝘍,[Γ@Ctx.Expression]⊢[𝛗@Fm.Expression])))",
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Henkin-Complete Extension F (Constructor: HI IpFau)\">𝘍</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-fau.html\" class=\"sym-link\" title=\"Non-r.e. Henkin-Complete Extension F (Constructor: HI IpFau)\">𝘍</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound context variable Γ\">Γ</a></span><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛗\">𝛗</a></span><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-fau.html\" class=\"sym-link\" title=\"Non-r.e. Henkin-Complete Extension F (Constructor: HI IpFau)\">𝘍</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound context variable Γ\">Γ</a></span><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛗\">𝛗</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/formula-metavariables.html}{\\Gamma} \\href{symbols/typing-colon.html}{:} \\href{symbols/ctx-sort.html}{\\mathsf{Ctx}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/typing-colon.html}{:} \\href{symbols/fm-sort.html}{\\mathsf{Fm}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-fau.html}{\\mathcal{F}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/quotation.html}{\\langle } \\href{symbols/theory-fau.html}{\\mathcal{F}} \\href{symbols/comma-delim.html}{,} \\href{symbols/formula-metavariables.html}{\\Gamma} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/quotation.html}{\\rangle } \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-fau.html}{\\mathcal{F}} \\href{symbols/comma-delim.html}{,} \\href{symbols/formula-metavariables.html}{\\Gamma} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
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      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "If a formula  is derived in 𝘍 from the empty context, then the formula 𝛗 is derived in 𝘍 from the context Γ. Why this holds for 𝘍: for an arbitrary theory 𝘞, concluding (𝘞, Γ⊢𝛗) from (𝘞, [] ⊢ ) requires 1-consistency / Σ₁-soundness of 𝘞 (and for the r.e. base theory 𝘛, asserting internal co-necessitation would by Löb's theorem 5.01 trivialize 𝘛). Because 𝘍 is a non-r.e. sound Henkin extension equipped with Soundness 3.73 ((𝘍, [] ⊢ ψ) → ⟦ψ⟧), instantiating 3.73 at ψ :=  directly yields (𝘍, Γ⊢𝛗).",
      "verification": "Primitive / Derived Rule (Kernel)"
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    {
      "index": 102,
      "name": "DefAtMostOneBinder",
      "id": "DefAtMostOneBinder",
      "legacy_id": "item-4.04",
      "num": "4.04",
      "title": "Quantifier “unique”(meta,WIP)",
      "kind": "Definition",
      "status": "Definitional (Kernel AST)",
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      "prf_constructor": "TpUniq",
      "scheme_of_types_unicode": "(∀([𝛗@Variable]:[Fm@Type])((!([z@Variable]:[Class@Type])⟦[𝛗@Fm.Expression]⟧)⟷(∀([x@Variable]:[Class@Type])∀([y@Variable]:[Class@Type])((⟦[𝛗@Fm.Expression][([x@Class.Expression.Expression]:Class.Expression)/‹z›]⟧∧(let z = [y@Class.Expression] in ⟦[𝛗@Fm.Expression]⟧))→([x@Class.Expression]=[y@Class.Expression])))))",
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href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/forall-binder.html\" class=\"sym-link\" title=\"Universal quantifier binder (Constructor: HP (TpFora ...))\">∀</a><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a></span><span class=\"ansi-light-yellow\"><a href=\"symbols/variables.html\" class=\"sym-link\" title=\"Bound class variable y (binder)\">y</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/typing-colon.html\" class=\"sym-link\" title=\"Typing judgment / conglomerate sort annotation colon (x:Class, A:Prop, T_e:Type)\">:</a></span><span class=\"ansi-bold-green\"><a href=\"symbols/class-type.html\" class=\"sym-link\" title=\"Default Type Class (universal conglomerate / constant-true predicate on classes)\">Class</a></span><span class=\"ansi-light-cyan\"><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/antiquotation.html\" class=\"sym-link\" title=\"Left semantic evaluation / 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      "index": 103,
      "name": "DefExistsUniqueBinder",
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      "index": 104,
      "name": "AxmMetaClassComprehension",
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ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-tau.html\" class=\"sym-link\" title=\"Recursively Enumerable Base Class Theory T (Constructor: HI IpTau)\">𝘛</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><span class=\"ansi-light-yellow\"><a href=\"symbols/formula-metavariables.html\" class=\"sym-link\" title=\"Bound formula variable 𝛗\">𝛗</a></span><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/forall-binder.html}{\\forall} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/typing-colon.html}{:} \\href{symbols/fm-sort.html}{\\mathsf{Fm}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-eau.html}{\\mathcal{E}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/quotation.html}{\\langle } \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/quotation.html}{\\langle } \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/antiquotation.html}{[\\![ } \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/antiquotation.html}{ ]\\!]} \\href{symbols/quotation.html}{\\rangle } \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/formula-metavariables.html}{\\boldsymbol{\\varphi}} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/quotation.html}{\\rangle } \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Here 𝘌 ∈ {IΣ₁, PA, 𝘛, NBG, MK} is an unspecified elementary metatheory capable of proving the Hilbert-Bernays-Löb derivability conditions D1-D3 for the r.e. base theory 𝘛 (specifically D3, Σ₁-completeness, which requires Σ₁-induction and fails in Robinson's arithmetic Q; see Hájek & Pudlák 1993, Metamathematics of First-Order Arithmetic, Ch. I §3 & Ch. III; Bezboruah & Shepherdson 1976, JSL 41(2):503-512; Boolos 1993, The Logic of Provability, Ch. 2).",
      "verification": "Verified (Kernel)"
    },
    {
      "index": 106,
      "name": "ThmSecondIncompleteness",
      "id": "ThmSecondIncompleteness",
      "legacy_id": "item-5.02",
      "num": "5.02",
      "title": "Gödel's second incompleteness theorem(WIP)",
      "kind": "Theorem",
      "status": "Verified (Kernel)",
      "formula_unicode": "(𝘌,[]⊢‹(¬(𝘛,[]⊢‹⊥›))→¬(𝘛,[]⊢‹¬(𝘛,[]⊢‹⊥›)›)›)",
      "prf_constructor": "TpHolds",
      "scheme_of_types_unicode": "(𝘌,[]⊢‹(¬(𝘛,[]⊢‹⊥›))→¬(𝘛,[]⊢‹¬(𝘛,[]⊢‹⊥›)›)›)",
      "kernel_ast_unicode": "(𝘌,[]⊢‹(¬(𝘛,[]⊢‹⊥›))→¬(𝘛,[]⊢‹¬(𝘛,[]⊢‹⊥›)›)›)",
      "formula_unicode_html": "<a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-eau.html\" class=\"sym-link\" title=\"Unspecified Elementary Metatheory E in {I\\Sigma_1, PA, T, NBG, MK}\">𝘌</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><a href=\"symbols/neg-conn.html\" class=\"sym-link\" title=\"Logical negation connective (Constructor: HP (TpNeg ...))\">¬</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-tau.html\" class=\"sym-link\" title=\"Recursively Enumerable Base Class Theory T (Constructor: HI IpTau)\">𝘛</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><span class=\"ansi-bold-blue\"><a href=\"symbols/bot-const.html\" class=\"sym-link\" title=\"Falsum / logical falsity constant (Constructor: HP TpBot)\">⊥</a></span><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/impl-conn.html\" class=\"sym-link\" title=\"Material implication connective / function arrow (Constructor: HP (TpImpl ...))\">→</a><a href=\"symbols/neg-conn.html\" class=\"sym-link\" title=\"Logical negation connective (Constructor: HP (TpNeg ...))\">¬</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-tau.html\" class=\"sym-link\" title=\"Recursively Enumerable Base Class Theory T (Constructor: HI IpTau)\">𝘛</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><a href=\"symbols/neg-conn.html\" class=\"sym-link\" title=\"Logical negation connective (Constructor: HP (TpNeg ...))\">¬</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Left grouping / binder domain / ordered pair / application parenthesis\">(</a><span class=\"ansi-light-magenta\"><a href=\"symbols/theory-tau.html\" class=\"sym-link\" title=\"Recursively Enumerable Base Class Theory T (Constructor: HI IpTau)\">𝘛</a></span><a href=\"symbols/comma-delim.html\" class=\"sym-link\" title=\"Tuple, sequent context, and binder body separator comma (,)\">,</a><a href=\"symbols/empty-list-const.html\" class=\"sym-link\" title=\"Empty Dunaev class-list / empty hypothesis context []\">[]</a><a href=\"symbols/turnstile-derivability.html\" class=\"sym-link\" title=\"Formal derivability turnstile predicate symbol (Constructor: HP (TpHolds (ApStmt ...)))\">⊢</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Left object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">‹</a><span class=\"ansi-bold-blue\"><a href=\"symbols/bot-const.html\" class=\"sym-link\" title=\"Falsum / logical falsity constant (Constructor: HP TpBot)\">⊥</a></span><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a><a href=\"symbols/quotation.html\" class=\"sym-link\" title=\"Right object-language syntactic quotation bracket <phi> (Constructor: HQuote)\">›</a><a href=\"symbols/delimiters-brackets.html\" class=\"sym-link\" title=\"Right grouping / binder domain / ordered pair / application parenthesis\">)</a>",
      "formula_latex_linked": "\\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-eau.html}{\\mathcal{E}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/quotation.html}{\\langle } \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/neg-conn.html}{\\neg} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/quotation.html}{\\langle } \\href{symbols/bot-const.html}{\\bot} \\href{symbols/quotation.html}{\\rangle } \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/impl-conn.html}{\\rightarrow} \\href{symbols/neg-conn.html}{\\neg} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/quotation.html}{\\langle } \\href{symbols/neg-conn.html}{\\neg} \\href{symbols/delimiters-brackets.html}{(} \\href{symbols/theory-tau.html}{\\mathcal{T}} \\href{symbols/comma-delim.html}{,} \\href{symbols/empty-list-const.html}{[\\,]} \\href{symbols/turnstile-derivability.html}{\\vdash} \\href{symbols/quotation.html}{\\langle } \\href{symbols/bot-const.html}{\\bot} \\href{symbols/quotation.html}{\\rangle } \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/quotation.html}{\\rangle } \\href{symbols/delimiters-brackets.html}{)} \\href{symbols/quotation.html}{\\rangle } \\href{symbols/delimiters-brackets.html}{)} ",
      "judgements_unicode": "",
      "judgements_unicode_html": "",
      "judgements_latex_linked": "",
      "side_condition_unicode": "",
      "side_condition_unicode_html": "",
      "side_condition_latex_linked": "",
      "comment": "Formalized in the unspecified elementary metatheory 𝘌 ∈ {IΣ₁, PA, 𝘛, NBG, MK} (not Robinson's Q, which lacks Σ₁-induction needed for HBL condition D3; cf. Hájek & Pudlák 1993; Bezboruah & Shepherdson 1976) for the recursively enumerable object theory 𝘛. Note that Gödel's second incompleteness theorem applies to the r.e. theory 𝘛, whereas 𝘍 is non-r.e.",
      "verification": "Verified (Kernel)"
    }
  ]
}
