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Theorem List for Metamath Proof Explorer - 35101-35200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1resrcmplf1d 35101 If a function's restriction to a subclass of its domain and its restriction to the relative complement of that subclass are both one-to-one, and if the ranges of those two restrictions are disjoint, then the function is itself one-to-one. (Contributed by BTernaryTau, 28-Sep-2023.)
(𝜑𝐶𝐴)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹𝐶):𝐶1-1𝐵)    &   (𝜑 → (𝐹 ↾ (𝐴𝐶)):(𝐴𝐶)–1-1𝐵)    &   (𝜑 → ((𝐹𝐶) ∩ (𝐹 “ (𝐴𝐶))) = ∅)       (𝜑𝐹:𝐴1-1𝐵)
 
Theoremfunen1cnv 35102 If a function is equinumerous to ordinal 1, then its converse is also a function. (Contributed by BTernaryTau, 8-Oct-2023.)
((Fun 𝐹𝐹 ≈ 1o) → Fun 𝐹)
 
Theoremfnrelpredd 35103* A function that preserves a relation also preserves predecessors. (Contributed by BTernaryTau, 16-Jul-2024.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)))    &   (𝜑𝐶𝐴)    &   (𝜑𝐷𝐴)       (𝜑 → Pred(𝑆, (𝐹𝐶), (𝐹𝐷)) = (𝐹 “ Pred(𝑅, 𝐶, 𝐷)))
 
Theoremcardpred 35104 The cardinality function preserves predecessors. (Contributed by BTernaryTau, 18-Jul-2024.)
((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))
 
Theoremnummin 35105* Every nonempty class of numerable sets has a minimal element. (Contributed by BTernaryTau, 18-Jul-2024.)
((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)
 
Theoremaxnulg 35106 A generalization of ax-nul 5306 in which 𝑥 and 𝑦 need not be distinct. Note that it is possible to use axc7e 2318 to derive elirrv 9636 from this theorem, which justifies the dependency on ax-reg 9632. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥
 
TheoremaxnulALT2 35107* Alternate proof of axnul 5305, proved from propositional calculus, ax-gen 1795, ax-4 1809, ax-5 1910, and ax-inf2 9681. (Contributed by BTernaryTau, 22-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥
 
21.5.2.1  Finitism
 
Theoremprcinf 35108* Any proper class is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. This proof holds regardless of whether the Axiom of Infinity is accepted or negated. (Contributed by BTernaryTau, 22-Jun-2025.)
𝐴 ∈ V → ∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛))
 
Theoremfineqvrep 35109* If the Axiom of Infinity is negated, then the Axiom of Replacement becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
(Fin = V → (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))))
 
Theoremfineqvpow 35110* If the Axiom of Infinity is negated, then the Axiom of Power Sets becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
(Fin = V → ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
 
Theoremfineqvac 35111 If the Axiom of Infinity is negated, then the Axiom of Choice becomes redundant. For a shorter proof using ax-rep 5279 and ax-pow 5365, see fineqvacALT 35112. (Contributed by BTernaryTau, 21-Sep-2024.)
(Fin = V → CHOICE)
 
TheoremfineqvacALT 35112 Shorter proof of fineqvac 35111 using ax-rep 5279 and ax-pow 5365. (Contributed by BTernaryTau, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fin = V → CHOICE)
 
21.5.2.2  Global choice
 
Theoremgblacfnacd 35113* If 𝐹 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10162) holds. Note that 𝐹 must be a proper class by fndmexb 7928. This means we cannot show that the existence of a class that behaves as a global choice function is sufficient because we only have existential quantifiers for sets, not (proper) classes. However, if a class variant of exlimiv 1930 were available, then it could be used alongside the closed form of this theorem to prove that result. (Contributed by BTernaryTau, 12-Dec-2024.)
(𝜑𝐹 Fn V)    &   (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))       (𝜑 → ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
 
Theoremwevgblacfn 35114* If 𝑅 is a well-ordering of the universe, then 𝐹 is a global choice function. Here 𝐹 maps each set 𝑧 to its minimal element with respect to 𝑅 (except when 𝑧 is the empty set, in which case it is mapped to the empty set, though this is only done for convenience). (Contributed by BTernaryTau, 29-Jun-2025.)
𝐹 = (𝑧 ∈ V ↦ {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦})       (𝑅 We V → (𝐹 Fn V ∧ ∀𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧)))
 
21.5.3  Real and complex numbers
 
Theoremzltp1ne 35115 Integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵𝐵 ≠ (𝐴 + 1))))
 
Theoremnnltp1ne 35116 Positive integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵𝐵 ≠ (𝐴 + 1))))
 
Theoremnn0ltp1ne 35117 Nonnegative integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵𝐵 ≠ (𝐴 + 1))))
 
Theorem0nn0m1nnn0 35118 A number is zero if and only if it's a nonnegative integer that becomes negative after subtracting 1. (Contributed by BTernaryTau, 30-Sep-2023.)
(𝑁 = 0 ↔ (𝑁 ∈ ℕ0 ∧ ¬ (𝑁 − 1) ∈ ℕ0))
 
Theoremf1resfz0f1d 35119 If a function with a sequence of nonnegative integers (starting at 0) as its domain is one-to-one when 0 is removed, and if the range of that restriction does not contain the function's value at the removed integer, then the function is itself one-to-one. (Contributed by BTernaryTau, 4-Oct-2023.)
(𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐹:(0...𝐾)⟶𝑉)    &   (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1𝑉)    &   (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)       (𝜑𝐹:(0...𝐾)–1-1𝑉)
 
Theoremfisshasheq 35120 A finite set is equal to its subset if they are the same size. (Contributed by BTernaryTau, 3-Oct-2023.)
((𝐵 ∈ Fin ∧ 𝐴𝐵 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝐴 = 𝐵)
 
Theoremrevpfxsfxrev 35121 The reverse of a prefix of a word is equal to the same-length suffix of the reverse of that word. (Contributed by BTernaryTau, 2-Dec-2023.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘(𝑊 prefix 𝐿)) = ((reverse‘𝑊) substr ⟨((♯‘𝑊) − 𝐿), (♯‘𝑊)⟩))
 
Theoremswrdrevpfx 35122 A subword expressed in terms of reverses and prefixes. (Contributed by BTernaryTau, 3-Dec-2023.)
((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝐹, 𝐿⟩) = (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))))
 
21.5.4  Graph theory
 
Theoremlfuhgr 35123* A hypergraph is loop-free if and only if every edge connects at least two vertices. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
 
Theoremlfuhgr2 35124* A hypergraph is loop-free if and only if every edge is not a loop. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1))
 
Theoremlfuhgr3 35125* A hypergraph is loop-free if and only if none of its edges connect to only one vertex. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ¬ ∃𝑎{𝑎} ∈ (Edg‘𝐺)))
 
Theoremcplgredgex 35126* Any two (distinct) vertices in a complete graph are connected to each other by at least one edge. (Contributed by BTernaryTau, 2-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ ComplGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
 
Theoremcusgredgex 35127 Any two (distinct) vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 3-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ ComplUSGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → {𝐴, 𝐵} ∈ 𝐸))
 
Theoremcusgredgex2 35128 Any two distinct vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 4-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ ComplUSGraph → ((𝐴𝑉𝐵𝑉𝐴𝐵) → {𝐴, 𝐵} ∈ 𝐸))
 
Theorempfxwlk 35129 A prefix of a walk is a walk. (Contributed by BTernaryTau, 2-Dec-2023.)
((𝐹(Walks‘𝐺)𝑃𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)))
 
Theoremrevwlk 35130 The reverse of a walk is a walk. (Contributed by BTernaryTau, 30-Nov-2023.)
(𝐹(Walks‘𝐺)𝑃 → (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃))
 
Theoremrevwlkb 35131 Two words represent a walk if and only if their reverses also represent a walk. (Contributed by BTernaryTau, 4-Dec-2023.)
((𝐹 ∈ Word 𝑊𝑃 ∈ Word 𝑈) → (𝐹(Walks‘𝐺)𝑃 ↔ (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃)))
 
Theoremswrdwlk 35132 Two matching subwords of a walk also represent a walk. (Contributed by BTernaryTau, 7-Dec-2023.)
((𝐹(Walks‘𝐺)𝑃𝐵 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 substr ⟨𝐵, 𝐿⟩)(Walks‘𝐺)(𝑃 substr ⟨𝐵, (𝐿 + 1)⟩))
 
Theorempthhashvtx 35133 A graph containing a path has at least as many vertices as there are edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.)
𝑉 = (Vtx‘𝐺)       (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉))
 
Theoremspthcycl 35134 A walk is a trivial path if and only if it is both a simple path and a cycle. (Contributed by BTernaryTau, 8-Oct-2023.)
((𝐹(Paths‘𝐺)𝑃𝐹 = ∅) ↔ (𝐹(SPaths‘𝐺)𝑃𝐹(Cycles‘𝐺)𝑃))
 
Theoremusgrgt2cycl 35135 A non-trivial cycle in a simple graph has a length greater than 2. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃𝐹 ≠ ∅) → 2 < (♯‘𝐹))
 
Theoremusgrcyclgt2v 35136 A simple graph with a non-trivial cycle must have at least 3 vertices. (Contributed by BTernaryTau, 5-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃𝐹 ≠ ∅) → 2 < (♯‘𝑉))
 
Theoremsubgrwlk 35137 If a walk exists in a subgraph of a graph 𝐺, then that walk also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)𝑃𝐹(Walks‘𝐺)𝑃))
 
Theoremsubgrtrl 35138 If a trail exists in a subgraph of a graph 𝐺, then that trail also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Trails‘𝑆)𝑃𝐹(Trails‘𝐺)𝑃))
 
Theoremsubgrpth 35139 If a path exists in a subgraph of a graph 𝐺, then that path also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Paths‘𝑆)𝑃𝐹(Paths‘𝐺)𝑃))
 
Theoremsubgrcycl 35140 If a cycle exists in a subgraph of a graph 𝐺, then that cycle also exists in 𝐺. (Contributed by BTernaryTau, 23-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Cycles‘𝑆)𝑃𝐹(Cycles‘𝐺)𝑃))
 
Theoremcusgr3cyclex 35141* Every complete simple graph with more than two vertices has a 3-cycle. (Contributed by BTernaryTau, 4-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
 
Theoremloop1cycl 35142* A hypergraph has a cycle of length one if and only if it has a loop. (Contributed by BTernaryTau, 13-Oct-2023.)
(𝐺 ∈ UHGraph → (∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ {𝐴} ∈ (Edg‘𝐺)))
 
Theorem2cycld 35143 Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)    &   (𝜑𝐴 = 𝐶)       (𝜑𝐹(Cycles‘𝐺)𝑃)
 
Theorem2cycl2d 35144 Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
𝑃 = ⟨“𝐴𝐵𝐴”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉))    &   (𝜑𝐴𝐵)    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐴, 𝐵} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)       (𝜑𝐹(Cycles‘𝐺)𝑃)
 
Theoremumgr2cycllem 35145* Lemma for umgr2cycl 35146. (Contributed by BTernaryTau, 17-Oct-2023.)
𝐹 = ⟨“𝐽𝐾”⟩    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UMGraph)    &   (𝜑𝐽 ∈ dom 𝐼)    &   (𝜑𝐽𝐾)    &   (𝜑 → (𝐼𝐽) = (𝐼𝐾))       (𝜑 → ∃𝑝 𝐹(Cycles‘𝐺)𝑝)
 
Theoremumgr2cycl 35146* A multigraph with two distinct edges that connect the same vertices has a 2-cycle. (Contributed by BTernaryTau, 17-Oct-2023.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UMGraph ∧ ∃𝑗 ∈ dom 𝐼𝑘 ∈ dom 𝐼((𝐼𝑗) = (𝐼𝑘) ∧ 𝑗𝑘)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 2))
 
21.5.4.1  Acyclic graphs
 
Syntaxcacycgr 35147 Extend class notation with acyclic graphs.
class AcyclicGraph
 
Definitiondf-acycgr 35148* Define the class of all acyclic graphs. A graph is called acyclic if it has no (non-trivial) cycles. (Contributed by BTernaryTau, 11-Oct-2023.)
AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅)}
 
Theoremdfacycgr1 35149* An alternate definition of the class of all acyclic graphs that requires all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.)
AcyclicGraph = {𝑔 ∣ ∀𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 = ∅)}
 
Theoremisacycgr 35150* The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
(𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
 
Theoremisacycgr1 35151* The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
(𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 = ∅)))
 
Theoremacycgrcycl 35152 Any cycle in an acyclic graph is trivial (i.e. has one vertex and no edges). (Contributed by BTernaryTau, 12-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → 𝐹 = ∅)
 
Theoremacycgr0v 35153 A null graph (with no vertices) is an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑊𝑉 = ∅) → 𝐺 ∈ AcyclicGraph)
 
Theoremacycgr1v 35154 A multigraph with one vertex is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1) → 𝐺 ∈ AcyclicGraph)
 
Theoremacycgr2v 35155 A simple graph with two vertices is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → 𝐺 ∈ AcyclicGraph)
 
Theoremprclisacycgr 35156* A proper class (representing a null graph, see vtxvalprc 29062) has the property of an acyclic graph (see also acycgr0v 35153). (Contributed by BTernaryTau, 11-Oct-2023.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)       𝐺 ∈ V → ¬ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅))
 
Theoremacycgrislfgr 35157* An acyclic hypergraph is a loop-free hypergraph. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
 
Theoremupgracycumgr 35158 An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, 15-Oct-2023.)
((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ UMGraph)
 
Theoremumgracycusgr 35159 An acyclic multigraph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph)
 
Theoremupgracycusgr 35160 An acyclic pseudograph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph)
 
Theoremcusgracyclt3v 35161 A complete simple graph is acyclic if and only if it has fewer than three vertices. (Contributed by BTernaryTau, 20-Oct-2023.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ (♯‘𝑉) < 3))
 
Theorempthacycspth 35162 A path in an acyclic graph is a simple path. (Contributed by BTernaryTau, 21-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝐹(Paths‘𝐺)𝑃) → 𝐹(SPaths‘𝐺)𝑃)
 
Theoremacycgrsubgr 35163 The subgraph of an acyclic graph is also acyclic. (Contributed by BTernaryTau, 23-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ AcyclicGraph)
 
21.6  Mathbox for Mario Carneiro
 
21.6.1  Predicate calculus with all distinct variables
 
Axiomax-7d 35164* Distinct variable version of ax-11 2157. (Contributed by Mario Carneiro, 14-Aug-2015.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Axiomax-8d 35165* Distinct variable version of ax-7 2007. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 
Axiomax-9d1 35166 Distinct variable version of ax-6 1967, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
¬ ∀𝑥 ¬ 𝑥 = 𝑥
 
Axiomax-9d2 35167* Distinct variable version of ax-6 1967, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
Axiomax-10d 35168* Distinct variable version of axc11n 2431. (Contributed by Mario Carneiro, 14-Aug-2015.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Axiomax-11d 35169* Distinct variable version of ax-12 2177. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
21.6.2  Miscellaneous stuff
 
Theoremquartfull 35170 The quartic equation, written out in full. This actually makes a fairly good Metamath stress test. Note that the length of this formula could be shortened significantly if the intermediate expressions were expanded and simplified, but it's not like this theorem will be used anyway. (Contributed by Mario Carneiro, 6-May-2015.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑 → (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)) ≠ 0)    &   (𝜑 → -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3) ≠ 0)       (𝜑 → ((((𝑋↑4) + (𝐴 · (𝑋↑3))) + ((𝐵 · (𝑋↑2)) + ((𝐶 · 𝑋) + 𝐷))) = 0 ↔ ((𝑋 = ((-(𝐴 / 4) − ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) + ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = ((-(𝐴 / 4) − ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) − (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) + ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)))))) ∨ (𝑋 = ((-(𝐴 / 4) + ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) − ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = ((-(𝐴 / 4) + ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) − (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) − ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)))))))))
 
21.6.3  Derangements and the Subfactorial
 
Theoremderanglem 35171* Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.)
(𝐴 ∈ Fin → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐴𝜑)} ∈ Fin)
 
Theoremderangval 35172* Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       (𝐴 ∈ Fin → (𝐷𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)}))
 
Theoremderangf 35173* The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       𝐷:Fin⟶ℕ0
 
Theoremderang0 35174* The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       (𝐷‘∅) = 1
 
Theoremderangsn 35175* The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       (𝐴𝑉 → (𝐷‘{𝐴}) = 0)
 
Theoremderangenlem 35176* One half of derangen 35177. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       ((𝐴𝐵𝐵 ∈ Fin) → (𝐷𝐴) ≤ (𝐷𝐵))
 
Theoremderangen 35177* The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       ((𝐴𝐵𝐵 ∈ Fin) → (𝐷𝐴) = (𝐷𝐵))
 
Theoremsubfacval 35178* The subfactorial is defined as the number of derangements (see derangval 35172) of the set (1...𝑁). (Contributed by Mario Carneiro, 21-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ0 → (𝑆𝑁) = (𝐷‘(1...𝑁)))
 
Theoremderangen2 35179* Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝐴 ∈ Fin → (𝐷𝐴) = (𝑆‘(♯‘𝐴)))
 
Theoremsubfacf 35180* The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       𝑆:ℕ0⟶ℕ0
 
Theoremsubfaclefac 35181* The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ0 → (𝑆𝑁) ≤ (!‘𝑁))
 
Theoremsubfac0 35182* The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑆‘0) = 1
 
Theoremsubfac1 35183* The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑆‘1) = 0
 
Theoremsubfacp1lem1 35184* Lemma for subfacp1 35191. The set 𝐾 together with {1, 𝑀} partitions the set 1...(𝑁 + 1). (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})       (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1)))
 
Theoremsubfacp1lem2a 35185* Lemma for subfacp1 35191. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐹 = (𝐺 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    &   (𝜑𝐺:𝐾1-1-onto𝐾)       (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹𝑀) = 1))
 
Theoremsubfacp1lem2b 35186* Lemma for subfacp1 35191. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐹 = (𝐺 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    &   (𝜑𝐺:𝐾1-1-onto𝐾)       ((𝜑𝑋𝐾) → (𝐹𝑋) = (𝐺𝑋))
 
Theoremsubfacp1lem3 35187* Lemma for subfacp1 35191. In subfacp1lem6 35190 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements that satisfy this for fixed 𝑀 = (𝑓‘1) is in bijection with 𝑁 − 1 derangements, by simply dropping the 𝑥 = 1 and 𝑥 = 𝑀 points from the function to get a derangement on 𝐾 = (1...(𝑁 − 1)) ∖ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐵 = {𝑔𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔𝑀) = 1)}    &   𝐶 = {𝑓 ∣ (𝑓:𝐾1-1-onto𝐾 ∧ ∀𝑦𝐾 (𝑓𝑦) ≠ 𝑦)}       (𝜑 → (♯‘𝐵) = (𝑆‘(𝑁 − 1)))
 
Theoremsubfacp1lem4 35188* Lemma for subfacp1 35191. The function 𝐹, which swaps 1 with 𝑀 and leaves all other elements alone, is a bijection of order 2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐵 = {𝑔𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔𝑀) ≠ 1)}    &   𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})       (𝜑𝐹 = 𝐹)
 
Theoremsubfacp1lem5 35189* Lemma for subfacp1 35191. In subfacp1lem6 35190 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements with (𝑓‘(𝑓‘1)) ≠ 1 for fixed 𝑀 = (𝑓‘1) is in bijection with derangements of 2...(𝑁 + 1), because pre-composing with the function 𝐹 swaps 1 and 𝑀 and turns the function into a bijection with (𝑓‘1) = 1 and (𝑓𝑥) ≠ 𝑥 for all other 𝑥, so dropping the point at 1 yields a derangement on the 𝑁 remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐵 = {𝑔𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔𝑀) ≠ 1)}    &   𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    &   𝐶 = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}       (𝜑 → (♯‘𝐵) = (𝑆𝑁))
 
Theoremsubfacp1lem6 35190* Lemma for subfacp1 35191. By induction, we cut up the set of all derangements on 𝑁 + 1 according to the 𝑁 possible values of (𝑓‘1) (since (𝑓‘1) ≠ 1), and for each set for fixed 𝑀 = (𝑓‘1), the subset of derangements with (𝑓𝑀) = 1 has size 𝑆(𝑁 − 1) (by subfacp1lem3 35187), while the subset with (𝑓𝑀) ≠ 1 has size 𝑆(𝑁) (by subfacp1lem5 35189). Adding it all up yields the desired equation 𝑁(𝑆(𝑁) + 𝑆(𝑁 − 1)) for the number of derangements on 𝑁 + 1. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}       (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
 
Theoremsubfacp1 35191* A two-term recurrence for the subfactorial. This theorem allows to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 35182, subfac1 35183. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))
 
Theoremsubfacval2 35192* A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ0 → (𝑆𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘))))
 
Theoremsubfaclim 35193* The subfactorial converges rapidly to 𝑁! / e. This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ → (abs‘(((!‘𝑁) / e) − (𝑆𝑁))) < (1 / 𝑁))
 
Theoremsubfacval3 35194* Another closed form expression for the subfactorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ → (𝑆𝑁) = (⌊‘(((!‘𝑁) / e) + (1 / 2))))
 
Theoremderangfmla 35195* The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (𝐷𝐴) = (⌊‘(((!‘(♯‘𝐴)) / e) + (1 / 2))))
 
21.6.4  The Erdős-Szekeres theorem
 
Theoremerdszelem1 35196* Lemma for erdsze 35207. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}       (𝑋𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋))
 
Theoremerdszelem2 35197* Lemma for erdsze 35207. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}       ((♯ “ 𝑆) ∈ Fin ∧ (♯ “ 𝑆) ⊆ ℕ)
 
Theoremerdszelem3 35198* Lemma for erdsze 35207. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))       (𝐴 ∈ (1...𝑁) → (𝐾𝐴) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}), ℝ, < ))
 
Theoremerdszelem4 35199* Lemma for erdsze 35207. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑂 Or ℝ       ((𝜑𝐴 ∈ (1...𝑁)) → {𝐴} ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)})
 
Theoremerdszelem5 35200* Lemma for erdsze 35207. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑂 Or ℝ       ((𝜑𝐴 ∈ (1...𝑁)) → (𝐾𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}))
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