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Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

See the MPE Most Recent Proofs page for news and some useful links.

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Last updated on 31-Mar-2026 at 7:17 AM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
26-Mar-2026gsumgfsumlem 16633 Shifting the indexes of a group sum indexed by consecutive integers. (Contributed by Jim Kingdon, 26-Mar-2026.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    &   𝑆 = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))    ⇒   (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝑆)))
 
26-Mar-2026gfsum0 16632 An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
(𝐺 ∈ CMnd → (𝐺 Σgf ∅) = (0g‘𝐺))
 
25-Mar-2026gsumgfsum 16634 On an integer range, Σg and Σgf agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σgf 𝐹))
 
25-Mar-2026gsumgfsum1 16631 On an integer range starting at one, Σg and Σgf agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐹:(1...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σgf 𝐹))
 
24-Mar-2026gfsumval 16630 Value of the finite group sum over an unordered finite set. (Contributed by Jim Kingdon, 24-Mar-2026.)
𝐵 = (Base‘𝑊)    &   (𝜑 → 𝑊 ∈ CMnd)    &   (𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴)    ⇒   (𝜑 → (𝑊 Σgf 𝐹) = (𝑊 Σg (𝐹 ∘ 𝐺)))
 
23-Mar-2026df-gfsum 16629 Define the finite group sum (iterated sum) over an unordered finite set. As currently defined, df-igsum 13335 is indexed by consecutive integers, but in the case of a commutative monoid, the order of the sum doesn't matter and we can define a sum indexed by any finite set without needing to specify an order. (Contributed by Jim Kingdon, 23-Mar-2026.)
Σgf = (𝑤 ∈ CMnd, 𝑓 ∈ V ↦ (℩𝑥(dom 𝑓 ∈ Fin ∧ ∃𝑔(𝑔:(1...(♯‘dom 𝑓))–1-1-onto→dom 𝑓 ∧ 𝑥 = (𝑤 Σg (𝑓 ∘ 𝑔))))))
 
20-Mar-2026exmidssfi 7125 Excluded middle is equivalent to any subset of a finite set being finite. Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 20-Mar-2026.)
(EXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin))
 
18-Mar-2026umgr1een 15969 A graph with one non-loop edge is a multigraph. (Contributed by Jim Kingdon, 18-Mar-2026.)
(𝜑 → 𝐾 ∈ 𝑋)    &   (𝜑 → 𝑉 ∈ 𝑌)    &   (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   (𝜑 → 𝐸 ≈ 2o)    ⇒   (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UMGraph)
 
18-Mar-2026upgr1een 15968 A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 15965 for a different way of specifying a graph with one edge. (Contributed by Jim Kingdon, 18-Mar-2026.)
(𝜑 → 𝐾 ∈ 𝑋)    &   (𝜑 → 𝑉 ∈ 𝑌)    &   (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   (𝜑 → 𝐸 ≈ 2o)    ⇒   (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)
 
14-Mar-2026trlsex 16196 The class of trails on a graph is a set. (Contributed by Jim Kingdon, 14-Mar-2026.)
(𝐺 ∈ 𝑉 → (Trails‘𝐺) ∈ V)
 
13-Mar-2026eupthv 16255 The classes involved in a Eulerian path are sets. (Contributed by Jim Kingdon, 13-Mar-2026.)
(𝐹(EulerPaths‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
 
13-Mar-20261hevtxdg0fi 16118 The vertex degree of vertex 𝐷 in a finite pseudograph 𝐺 with only one edge 𝐸 is 0 if 𝐷 is not incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.) (Revised by Jim Kingdon, 13-Mar-2026.)
(𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   (𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑋)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝐺 ∈ UPGraph)    &   (𝜑 → 𝐸 ∈ 𝑌)    &   (𝜑 → 𝐷 ∉ 𝐸)    ⇒   (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 0)
 
11-Mar-2026en1hash 11055 A set equinumerous to the ordinal one has size 1 . (Contributed by Jim Kingdon, 11-Mar-2026.)
(𝐴 ≈ 1o → (♯‘𝐴) = 1)
 
4-Mar-2026elmpom 6398 If a maps-to operation is inhabited, the first class it is defined with is inhabited. (Contributed by Jim Kingdon, 4-Mar-2026.)
𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   (𝐷 ∈ 𝐹 → ∃𝑧 𝑧 ∈ 𝐴)
 
22-Feb-2026isclwwlkni 16216 A word over the set of vertices representing a closed walk of a fixed length. (Contributed by Jim Kingdon, 22-Feb-2026.)
(𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))
 
21-Feb-2026clwwlkex 16207 Existence of the set of closed walks (represented by words). (Contributed by Jim Kingdon, 21-Feb-2026.)
(𝐺 ∈ 𝑉 → (ClWWalks‘𝐺) ∈ V)
 
17-Feb-2026vtxdgfif 16104 In a finite graph, the vertex degree function is a function from vertices to nonnegative integers. (Contributed by Jim Kingdon, 17-Feb-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝐺 ∈ UPGraph)    ⇒   (𝜑 → (VtxDeg‘𝐺):𝑉⟶ℕ0)
 
16-Feb-2026vtxlpfi 16101 In a finite graph, the number of loops from a given vertex is finite. (Contributed by Jim Kingdon, 16-Feb-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ UPGraph)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} ∈ Fin)
 
16-Feb-2026vtxedgfi 16100 In a finite graph, the number of edges from a given vertex is finite. (Contributed by Jim Kingdon, 16-Feb-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ UPGraph)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)} ∈ Fin)
 
15-Feb-2026eqsndc 7090 Decidability of equality between a finite subset of a set with decidable equality, and a singleton whose element is an element of the larger set. (Contributed by Jim Kingdon, 15-Feb-2026.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → DECID 𝐴 = {𝑋})
 
14-Feb-2026pw1ninf 16540 The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7097), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7093. (Contributed by Jim Kingdon, 14-Feb-2026.)
¬ ω ≼ 𝒫 1o
 
14-Feb-2026pw1ndom3 16539 The powerset of 1o does not dominate 3o. This is another way of saying that 𝒫 1o does not have three elements (like pwntru 4287). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)
¬ 3o ≼ 𝒫 1o
 
14-Feb-2026pw1ndom3lem 16538 Lemma for pw1ndom3 16539. (Contributed by Jim Kingdon, 14-Feb-2026.)
(𝜑 → 𝑋 ∈ 𝒫 1o)    &   (𝜑 → 𝑌 ∈ 𝒫 1o)    &   (𝜑 → 𝑍 ∈ 𝒫 1o)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑋 ≠ 𝑍)    &   (𝜑 → 𝑌 ≠ 𝑍)    ⇒   (𝜑 → 𝑋 = ∅)
 
12-Feb-2026pw1dceq 16555 The powerset of 1o having decidable equality is equivalent to excluded middle. (Contributed by Jim Kingdon, 12-Feb-2026.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
 
12-Feb-20263dom 16537 A set that dominates ordinal 3 has at least 3 different members. (Contributed by Jim Kingdon, 12-Feb-2026.)
(3o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧))
 
11-Feb-2026elssdc 7089 Membership in a finite subset of a set with decidable equality is decidable. (Contributed by Jim Kingdon, 11-Feb-2026.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → DECID 𝑋 ∈ 𝐴)
 
10-Feb-2026vtxdgfifival 16102 The degree of a vertex for graphs with finite vertex and edge sets. (Contributed by Jim Kingdon, 10-Feb-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ UPGraph)    ⇒   (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
 
10-Feb-2026fidcen 7083 Equinumerosity of finite sets is decidable. (Contributed by Jim Kingdon, 10-Feb-2026.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → DECID 𝐴 ≈ 𝐵)
 
8-Feb-2026wlkvtxm 16151 A graph with a walk has at least one vertex. (Contributed by Jim Kingdon, 8-Feb-2026.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝐹(Walks‘𝐺)𝑃 → ∃𝑥 𝑥 ∈ 𝑉)
 
7-Feb-2026trlsv 16193 The classes involved in a trail are sets. (Contributed by Jim Kingdon, 7-Feb-2026.)
(𝐹(Trails‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
 
7-Feb-2026wlkex 16136 The class of walks on a graph is a set. (Contributed by Jim Kingdon, 7-Feb-2026.)
(𝐺 ∈ 𝑉 → (Walks‘𝐺) ∈ V)
 
3-Feb-2026dom1oi 6998 A set with an element dominates one. (Contributed by Jim Kingdon, 3-Feb-2026.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 1o ≼ 𝐴)
 
2-Feb-2026edginwlkd 16166 The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.) (Revised by Jim Kingdon, 2-Feb-2026.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → 𝐹 ∈ Word dom 𝐼)    &   (𝜑 → 𝐾 ∈ (0..^(♯‘𝐹)))    &   (𝜑 → 𝐺 ∈ 𝑉)    ⇒   (𝜑 → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸)
 
2-Feb-2026wlkelvv 16160 A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)
(𝑊 ∈ (Walks‘𝐺) → 𝑊 ∈ (V × V))
 
1-Feb-2026wlkcprim 16161 A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Revised by Jim Kingdon, 1-Feb-2026.)
(𝑊 ∈ (Walks‘𝐺) → (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊))
 
1-Feb-2026wlkmex 16130 If there are walks on a graph, the graph is a set. (Contributed by Jim Kingdon, 1-Feb-2026.)
(𝑊 ∈ (Walks‘𝐺) → 𝐺 ∈ V)
 
31-Jan-2026fvmbr 5670 If a function value is inhabited, the argument is related to the function value. (Contributed by Jim Kingdon, 31-Jan-2026.)
(𝐴 ∈ (𝐹‘𝑋) → 𝑋𝐹(𝐹‘𝑋))
 
30-Jan-2026elfvfvex 5669 If a function value is inhabited, the function value is a set. (Contributed by Jim Kingdon, 30-Jan-2026.)
(𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ∈ V)
 
30-Jan-2026reldmm 4948 A relation is inhabited iff its domain is inhabited. (Contributed by Jim Kingdon, 30-Jan-2026.)
(Rel 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ dom 𝐴))
 
25-Jan-2026ifp2 986 Forward direction of dfifp2dc 987. This direction does not require decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
(if-(𝜑, 𝜓, 𝜒) → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
 
25-Jan-2026ifpdc 985 The conditional operator for propositions implies decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
(if-(𝜑, 𝜓, 𝜒) → DECID 𝜑)
 
20-Jan-2026cats1fvd 11340 A symbol other than the last in a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 20-Jan-2026.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑 → 𝑆 ∈ Word V)    &   (𝜑 → (♯‘𝑆) = 𝑀)    &   (𝜑 → 𝑌 ∈ 𝑉)    &   (𝜑 → 𝑋 ∈ 𝑊)    &   (𝜑 → (𝑆‘𝑁) = 𝑌)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑁 < 𝑀)    ⇒   (𝜑 → (𝑇‘𝑁) = 𝑌)
 
20-Jan-2026cats1fvnd 11339 The last symbol of a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 20-Jan-2026.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑 → 𝑆 ∈ Word V)    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → (♯‘𝑆) = 𝑀)    ⇒   (𝜑 → (𝑇‘𝑀) = 𝑋)
 
19-Jan-2026cats2catd 11343 Closure of concatenation of concatenations with singleton words. (Contributed by AV, 1-Mar-2021.) (Revised by Jim Kingdon, 19-Jan-2026.)
(𝜑 → 𝐵 ∈ Word V)    &   (𝜑 → 𝐷 ∈ Word V)    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → 𝑌 ∈ 𝑊)    &   (𝜑 → 𝐴 = (𝐵 ++ ⟨“𝑋”⟩))    &   (𝜑 → 𝐶 = (⟨“𝑌”⟩ ++ 𝐷))    ⇒   (𝜑 → (𝐴 ++ 𝐶) = ((𝐵 ++ ⟨“𝑋𝑌”⟩) ++ 𝐷))
 
19-Jan-2026cats1catd 11342 Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 19-Jan-2026.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑 → 𝐴 ∈ Word V)    &   (𝜑 → 𝑆 ∈ Word V)    &   (𝜑 → 𝑋 ∈ 𝑊)    &   (𝜑 → 𝐶 = (𝐵 ++ ⟨“𝑋”⟩))    &   (𝜑 → 𝐵 = (𝐴 ++ 𝑆))    ⇒   (𝜑 → 𝐶 = (𝐴 ++ 𝑇))
 
19-Jan-2026cats1lend 11341 The length of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 19-Jan-2026.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑 → 𝑆 ∈ Word V)    &   (𝜑 → 𝑋 ∈ 𝑊)    &   (♯‘𝑆) = 𝑀    &   (𝑀 + 1) = 𝑁    ⇒   (𝜑 → (♯‘𝑇) = 𝑁)
 
18-Jan-2026rexanaliim 2636 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Revised by Jim Kingdon, 18-Jan-2026.)
(∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
 
15-Jan-2026df-uspgren 15999 Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph or a special undirected simple hypergraph, consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") into subsets of 𝑣 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a pseudograph, there is at most one edge between two vertices resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by Jim Kingdon, 15-Jan-2026.)
USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}}
 
11-Jan-2026en2prde 7392 A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by Jim Kingdon, 11-Jan-2026.)
(𝑉 ≈ 2o → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
 
10-Jan-2026pw1mapen 16547 Equinumerosity of (𝒫 1o ↑𝑚 𝐴) and the set of subsets of 𝐴. (Contributed by Jim Kingdon, 10-Jan-2026.)
(𝐴 ∈ 𝑉 → (𝒫 1o ↑𝑚 𝐴) ≈ 𝒫 𝐴)
 
10-Jan-2026pw1if 7436 Expressing a truth value in terms of an if expression. (Contributed by Jim Kingdon, 10-Jan-2026.)
(𝐴 ∈ 𝒫 1o → if(𝐴 = 1o, 1o, ∅) = 𝐴)
 
10-Jan-2026pw1m 7435 A truth value which is inhabited is equal to true. This is a variation of pwntru 4287 and pwtrufal 16548. (Contributed by Jim Kingdon, 10-Jan-2026.)
((𝐴 ∈ 𝒫 1o ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 = 1o)
 
10-Jan-20261ndom2 7046 Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
¬ 2o ≼ 1o
 
9-Jan-2026pw1map 16546 Mapping between (𝒫 1o ↑𝑚 𝐴) and subsets of 𝐴. (Contributed by Jim Kingdon, 9-Jan-2026.)
𝐹 = (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})    ⇒   (𝐴 ∈ 𝑉 → 𝐹:(𝒫 1o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)
 
9-Jan-2026iftrueb01 7434 Using an if expression to represent a truth value by ∅ or 1o. Unlike some theorems using if, 𝜑 does not need to be decidable. (Contributed by Jim Kingdon, 9-Jan-2026.)
(if(𝜑, 1o, ∅) = 1o ↔ 𝜑)
 
8-Jan-2026pfxclz 11253 Closure of the prefix extractor. This extends pfxclg 11252 from ℕ0 to ℤ (negative lengths are trivial, resulting in the empty word). (Contributed by Jim Kingdon, 8-Jan-2026.)
((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℤ) → (𝑆 prefix 𝐿) ∈ Word 𝐴)
 
8-Jan-2026fnpfx 11251 The domain of the prefix extractor. (Contributed by Jim Kingdon, 8-Jan-2026.)
prefix Fn (V × ℕ0)
 
7-Jan-2026pr1or2 7393 An unordered pair, with decidable equality for the specified elements, has either one or two elements. (Contributed by Jim Kingdon, 7-Jan-2026.)
((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ DECID 𝐴 = 𝐵) → ({𝐴, 𝐵} ≈ 1o ∨ {𝐴, 𝐵} ≈ 2o))
 
6-Jan-2026upgr1elem1 15964 Lemma for upgr1edc 15965. (Contributed by AV, 16-Oct-2020.) (Revised by Jim Kingdon, 6-Jan-2026.)
(𝜑 → {𝐵, 𝐶} ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → DECID 𝐵 = 𝐶)    ⇒   (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ 𝑆 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
 
3-Jan-2026df-umgren 15938 Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to a pseudograph, a multigraph has no loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." (Contributed by AV, 24-Nov-2020.) (Revised by Jim Kingdon, 3-Jan-2026.)
UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}}
 
3-Jan-2026df-upgren 15937 Define the class of all undirected pseudographs. An (undirected) pseudograph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop." (in contrast to a multigraph, see df-umgren 15938). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.) (Revised by Jim Kingdon, 3-Jan-2026.)
UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}}
 
3-Jan-2026dom1o 6997 Two ways of saying that a set is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ∈ 𝑉 → (1o ≼ 𝐴 ↔ ∃𝑗 𝑗 ∈ 𝐴))
 
3-Jan-2026en2m 6994 A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ≈ 2o → ∃𝑥 𝑥 ∈ 𝐴)
 
3-Jan-2026en1m 6974 A set with one element is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ≈ 1o → ∃𝑥 𝑥 ∈ 𝐴)
 
31-Dec-2025pw0ss 15927 There are no inhabited subsets of the empty set. (Contributed by Jim Kingdon, 31-Dec-2025.)
{𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅
 
31-Dec-2025df-ushgrm 15914 Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function 𝑒 is an injective (one-to-one) function into subsets of the set of vertices 𝑣, representing the (one or more) vertices incident to the edge. This definition corresponds to the definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subset of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are nonempty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by Jim Kingdon, 31-Dec-2025.)
USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}}
 
29-Dec-2025df-uhgrm 15913 Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into the set of inhabited subsets of this set. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by Jim Kingdon, 29-Dec-2025.)
UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}}
 
29-Dec-2025iedgex 15863 Applying the indexed edge function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (iEdg‘𝐺) ∈ V)
 
29-Dec-2025vtxex 15862 Applying the vertex function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (Vtx‘𝐺) ∈ V)
 
29-Dec-2025snmb 3791 A singleton is inhabited iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.) (Revised by Jim Kingdon, 29-Dec-2025.)
(𝐴 ∈ V ↔ ∃𝑥 𝑥 ∈ {𝐴})
 
27-Dec-2025lswex 11158 Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11155 or lswcl 11157 if you want more specific results for empty or nonempty words, respectively. (Contributed by Jim Kingdon, 27-Dec-2025.)
(𝑊 ∈ Word 𝑉 → (lastS‘𝑊) ∈ V)
 
23-Dec-2025fzowrddc 11221 Decidability of whether a range of integers is a subset of a word's domain. (Contributed by Jim Kingdon, 23-Dec-2025.)
((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → DECID (𝐹..^𝐿) ⊆ dom 𝑆)
 
19-Dec-2025ccatclab 11164 The concatenation of words over two sets is a word over the union of those sets. (Contributed by Jim Kingdon, 19-Dec-2025.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word (𝐴 ∪ 𝐵))
 
18-Dec-2025lswwrd 11153 Extract the last symbol of a word. (Contributed by Alexander van der Vekens, 18-Mar-2018.) (Revised by Jim Kingdon, 18-Dec-2025.)
(𝑊 ∈ Word 𝑉 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
 
14-Dec-20252strstrndx 13194 A constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 14-Dec-2025.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩)
 
12-Dec-2025funiedgdm2vald 15876 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 12-Dec-2025.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝜑 → 𝐺 ∈ 𝑋)    &   (𝜑 → Fun (𝐺 ∖ {∅}))    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → {𝐴, 𝐵} ⊆ dom 𝐺)    ⇒   (𝜑 → (iEdg‘𝐺) = (.ef‘𝐺))
 
11-Dec-2025funvtxdm2vald 15875 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝜑 → 𝐺 ∈ 𝑋)    &   (𝜑 → Fun (𝐺 ∖ {∅}))    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → {𝐴, 𝐵} ⊆ dom 𝐺)    ⇒   (𝜑 → (Vtx‘𝐺) = (Base‘𝐺))
 
11-Dec-2025funiedgdm2domval 15874 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2o ≼ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))
 
11-Dec-2025funvtxdm2domval 15873 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2o ≼ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))
 
4-Dec-2025hash2en 11100 Two equivalent ways to say a set has two elements. (Contributed by Jim Kingdon, 4-Dec-2025.)
(𝑉 ≈ 2o ↔ (𝑉 ∈ Fin ∧ (♯‘𝑉) = 2))
 
30-Nov-2025nninfnfiinf 16575 An element of ℕ∞ which is not finite is infinite. (Contributed by Jim Kingdon, 30-Nov-2025.)
((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → 𝐴 = (𝑖 ∈ ω ↦ 1o))
 
30-Nov-2025eluz3nn 9794 An integer greater than or equal to 3 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.) (Proof shortened by AV, 30-Nov-2025.)
(𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
 
27-Nov-2025psrelbasfi 14683 Simpler form of psrelbas 14682 when the index set is finite. (Contributed by Jim Kingdon, 27-Nov-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐾 = (Base‘𝑅)    &   (𝜑 → 𝐼 ∈ Fin)    &   𝐵 = (Base‘𝑆)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑋:(ℕ0 ↑𝑚 𝐼)⟶𝐾)
 
26-Nov-2025mplsubgfileminv 14707 Lemma for mplsubgfi 14708. The additive inverse of a polynomial is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝑈)    &   𝑁 = (invg‘𝑆)    ⇒   (𝜑 → (𝑁‘𝑋) ∈ 𝑈)
 
26-Nov-2025mplsubgfilemcl 14706 Lemma for mplsubgfi 14708. The sum of two polynomials is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝑈)    &   (𝜑 → 𝑌 ∈ 𝑈)    &    + = (+g‘𝑆)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)
 
25-Nov-2025nninfinfwlpo 7373 The point at infinity in ℕ∞ being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO). By isolated, we mean that the equality of that point with every other element of ℕ∞ is decidable. From an online post by Martin Escardo. By contrast, elements of ℕ∞ corresponding to natural numbers are isolated (nninfisol 7326). (Contributed by Jim Kingdon, 25-Nov-2025.)
(∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ω ∈ WOmni)
 
23-Nov-2025psrbagfi 14680 A finite index set gives a simpler expression for finite bags. (Contributed by Jim Kingdon, 23-Nov-2025.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   (𝐼 ∈ Fin → 𝐷 = (ℕ0 ↑𝑚 𝐼))
 
22-Nov-2025df-acnm 7378 Define a local and length-limited version of the axiom of choice. The definition of the predicate 𝑋 ∈ AC 𝐴 is that for all families of inhabited subsets of 𝑋 indexed on 𝐴 (i.e. functions 𝐴⟶{𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗𝑗 ∈ 𝑧}), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.) Change nonempty to inhabited. (Revised by Jim Kingdon, 22-Nov-2025.)
AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
 
21-Nov-2025mplsubgfilemm 14705 Lemma for mplsubgfi 14708. There exists a polynomial. (Contributed by Jim Kingdon, 21-Nov-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Grp)    ⇒   (𝜑 → ∃𝑗 𝑗 ∈ 𝑈)
 
15-Nov-2025uzuzle35 9792 An integer greater than or equal to 5 is an integer greater than or equal to 3. (Contributed by AV, 15-Nov-2025.)
(𝐴 ∈ (ℤ≥‘5) → 𝐴 ∈ (ℤ≥‘3))
 
14-Nov-20252omapen 16545 Equinumerosity of (2o ↑𝑚 𝐴) and the set of decidable subsets of 𝐴. (Contributed by Jim Kingdon, 14-Nov-2025.)
(𝐴 ∈ 𝑉 → (2o ↑𝑚 𝐴) ≈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥})
 
12-Nov-20252omap 16544 Mapping between (2o ↑𝑚 𝐴) and decidable subsets of 𝐴. (Contributed by Jim Kingdon, 12-Nov-2025.)
𝐹 = (𝑠 ∈ (2o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})    ⇒   (𝐴 ∈ 𝑉 → 𝐹:(2o ↑𝑚 𝐴)–1-1-onto→{𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥})
 
11-Nov-2025domomsubct 16552 A set dominated by ω is subcountable. (Contributed by Jim Kingdon, 11-Nov-2025.)
(𝐴 ≼ ω → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴))
 
10-Nov-2025prdsbaslemss 13350 Lemma for prdsbas 13352 and similar theorems. (Contributed by Jim Kingdon, 10-Nov-2025.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐴 = (𝐸‘𝑃)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   (𝜑 → 𝑇 ∈ 𝑋)    &   (𝜑 → {⟨(𝐸‘ndx), 𝑇⟩} ⊆ 𝑃)    ⇒   (𝜑 → 𝐴 = 𝑇)
 
5-Nov-2025fnmpl 14700 mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
mPoly Fn (V × V)
 
4-Nov-2025mplelbascoe 14699 Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g‘𝑅)    &   𝑈 = (Base‘𝑃)    ⇒   ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑋‘𝑏) = 0 ))))
 
4-Nov-2025mplbascoe 14698 Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g‘𝑅)    &   𝑈 = (Base‘𝑃)    ⇒   ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑈 = {𝑓 ∈ 𝐵 ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )})
 
4-Nov-2025mplvalcoe 14697 Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g‘𝑅)    &   𝑈 = {𝑓 ∈ 𝐵 ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )}    ⇒   ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑃 = (𝑆 ↾s 𝑈))
 
1-Nov-2025ficardon 7387 The cardinal number of a finite set is an ordinal. (Contributed by Jim Kingdon, 1-Nov-2025.)
(𝐴 ∈ Fin → (card‘𝐴) ∈ On)
 
31-Oct-2025bitsdc 12501 Whether a bit is set is decidable. (Contributed by Jim Kingdon, 31-Oct-2025.)
((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → DECID 𝑀 ∈ (bits‘𝑁))
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