Most Recent Proofs - Intuitionistic Logic Explorer

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.

Last updated on 19-Mar-2026 at 7:05 AM ET.

Recent Additions to the Intuitionistic Logic Explorer

Date	Label	Description
Theorem
14-Mar-2026	trlsex 16182	The class of trails on a graph is a set. (Contributed by Jim Kingdon, 14-Mar-2026.)
|- ( G e. V -> (Trails ` G ) e. _V )
13-Mar-2026	eupthv 16241	The classes involved in a Eulerian path are sets. (Contributed by Jim Kingdon, 13-Mar-2026.)
|- ( F (EulerPaths ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
13-Mar-2026	1hevtxdg0fi 16113	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.)
|- ( ph -> (iEdg ` G ) = { <. A , E >. } )   &    |- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> A e. X )   &    |- ( ph -> D e. V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> E e. Y )   &    |- ( ph -> D e/ E )   =>    |- ( ph -> ( (VtxDeg ` G ) ` D ) = 0 )
11-Mar-2026	en1hash 11052	A set equinumerous to the ordinal one has size 1 . (Contributed by Jim Kingdon, 11-Mar-2026.)
|- ( A ~~ 1o -> (♯ ` A ) = 1 )
4-Mar-2026	elmpom 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 e. A , y e. B |-> C )   =>    |- ( D e. F -> E. z z e. A )
22-Feb-2026	isclwwlkni 16202	A word over the set of vertices representing a closed walk of a fixed length. (Contributed by Jim Kingdon, 22-Feb-2026.)
|- ( W e. ( N ClWWalksN G ) -> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) )
21-Feb-2026	clwwlkex 16193	Existence of the set of closed walks (represented by words). (Contributed by Jim Kingdon, 21-Feb-2026.)
|- ( G e. V -> (ClWWalks ` G ) e. _V )
17-Feb-2026	vtxdgfif 16099	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   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> (VtxDeg ` G ) : V --> NN0 )
16-Feb-2026	vtxlpfi 16096	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   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> { x e. A | ( I ` x ) = { U } } e. Fin )
16-Feb-2026	vtxedgfi 16095	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   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> { x e. A | U e. ( I ` x ) } e. Fin )
15-Feb-2026	eqsndc 7088	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.)
|- ( ph -> A. x e. B A. y e. B DECID x = y )   &    |- ( ph -> X e. B )   &    |- ( ph -> A C_ B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> DECID A = { X } )
14-Feb-2026	pw1ninf 16526	The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7095), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7091. (Contributed by Jim Kingdon, 14-Feb-2026.)
|- -. om ~<_ ~P 1o
14-Feb-2026	pw1ndom3 16525	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-2026	pw1ndom3lem 16524	Lemma for pw1ndom3 16525. (Contributed by Jim Kingdon, 14-Feb-2026.)
|- ( ph -> X e. ~P 1o )   &    |- ( ph -> Y e. ~P 1o )   &    |- ( ph -> Z e. ~P 1o )   &    |- ( ph -> X =/= Y )   &    |- ( ph -> X =/= Z )   &    |- ( ph -> Y =/= Z )   =>    |- ( ph -> X = (/) )
12-Feb-2026	pw1dceq 16541	The powerset of 1o having decidable equality is equivalent to excluded middle. (Contributed by Jim Kingdon, 12-Feb-2026.)
|- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1oDECID x = y )
12-Feb-2026	3dom 16523	A set that dominates ordinal 3 has at least 3 different members. (Contributed by Jim Kingdon, 12-Feb-2026.)
|- ( 3o ~<_ A -> E. x e. A E. y e. A E. z e. A ( x =/= y /\ x =/= z /\ y =/= z ) )
11-Feb-2026	elssdc 7087	Membership in a finite subset of a set with decidable equality is decidable. (Contributed by Jim Kingdon, 11-Feb-2026.)
|- ( ph -> A. x e. B A. y e. B DECID x = y )   &    |- ( ph -> X e. B )   &    |- ( ph -> A C_ B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> DECID X e. A )
10-Feb-2026	vtxdgfifival 16097	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   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> ( (VtxDeg ` G ) ` U ) = ( (♯ ` { x e. A | U e. ( I ` x ) } ) + (♯ ` { x e. A | ( I ` x ) = { U } } ) ) )
10-Feb-2026	fidcen 7081	Equinumerosity of finite sets is decidable. (Contributed by Jim Kingdon, 10-Feb-2026.)
|- ( ( A e. Fin /\ B e. Fin ) -> DECID A ~~ B )
8-Feb-2026	wlkvtxm 16137	A graph with a walk has at least one vertex. (Contributed by Jim Kingdon, 8-Feb-2026.)
|- V = (Vtx ` G )   =>    |- ( F (Walks ` G ) P -> E. x x e. V )
7-Feb-2026	trlsv 16179	The classes involved in a trail are sets. (Contributed by Jim Kingdon, 7-Feb-2026.)
|- ( F (Trails ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
7-Feb-2026	wlkex 16122	The class of walks on a graph is a set. (Contributed by Jim Kingdon, 7-Feb-2026.)
|- ( G e. V -> (Walks ` G ) e. _V )
3-Feb-2026	dom1oi 6998	A set with an element dominates one. (Contributed by Jim Kingdon, 3-Feb-2026.)
|- ( ( A e. V /\ B e. A ) -> 1o ~<_ A )
2-Feb-2026	edginwlkd 16152	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 )   &    |- ( ph -> Fun I )   &    |- ( ph -> F e. Word dom I )   &    |- ( ph -> K e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> G e. V )   =>    |- ( ph -> ( I ` ( F ` K ) ) e. E )
2-Feb-2026	wlkelvv 16146	A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)
|- ( W e. (Walks ` G ) -> W e. ( _V X. _V ) )
1-Feb-2026	wlkcprim 16147	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 e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )
1-Feb-2026	wlkmex 16116	If there are walks on a graph, the graph is a set. (Contributed by Jim Kingdon, 1-Feb-2026.)
|- ( W e. (Walks ` G ) -> G e. _V )
31-Jan-2026	fvmbr 5670	If a function value is inhabited, the argument is related to the function value. (Contributed by Jim Kingdon, 31-Jan-2026.)
|- ( A e. ( F ` X ) -> X F ( F ` X ) )
30-Jan-2026	elfvfvex 5669	If a function value is inhabited, the function value is a set. (Contributed by Jim Kingdon, 30-Jan-2026.)
|- ( A e. ( F ` B ) -> ( F ` B ) e. _V )
30-Jan-2026	reldmm 4948	A relation is inhabited iff its domain is inhabited. (Contributed by Jim Kingdon, 30-Jan-2026.)
|- ( Rel A -> ( E. x x e. A <-> E. y y e. dom A ) )
25-Jan-2026	ifp2 986	Forward direction of dfifp2dc 987. This direction does not require decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
|- (if- ( ph , ps , ch ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
25-Jan-2026	ifpdc 985	The conditional operator for propositions implies decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
|- (if- ( ph , ps , ch ) -> DECID ph )
20-Jan-2026	cats1fvd 11337	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 "> )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> (♯ ` S ) = M )   &    |- ( ph -> Y e. V )   &    |- ( ph -> X e. W )   &    |- ( ph -> ( S ` N ) = Y )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> N < M )   =>    |- ( ph -> ( T ` N ) = Y )
20-Jan-2026	cats1fvnd 11336	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 "> )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> X e. V )   &    |- ( ph -> (♯ ` S ) = M )   =>    |- ( ph -> ( T ` M ) = X )
19-Jan-2026	cats2catd 11340	Closure of concatenation of concatenations with singleton words. (Contributed by AV, 1-Mar-2021.) (Revised by Jim Kingdon, 19-Jan-2026.)
|- ( ph -> B e. Word _V )   &    |- ( ph -> D e. Word _V )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. W )   &    |- ( ph -> A = ( B ++ <" X "> ) )   &    |- ( ph -> C = ( <" Y "> ++ D ) )   =>    |- ( ph -> ( A ++ C ) = ( ( B ++ <" X Y "> ) ++ D ) )
19-Jan-2026	cats1catd 11339	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 19-Jan-2026.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> A e. Word _V )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> X e. W )   &    |- ( ph -> C = ( B ++ <" X "> ) )   &    |- ( ph -> B = ( A ++ S ) )   =>    |- ( ph -> C = ( A ++ T ) )
19-Jan-2026	cats1lend 11338	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 "> )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> X e. W )   &    |- (♯ ` S ) = M   &    |- ( M + 1 ) = N   =>    |- ( ph -> (♯ ` T ) = N )
18-Jan-2026	rexanaliim 2636	A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Revised by Jim Kingdon, 18-Jan-2026.)
|- ( E. x e. A ( ph /\ -. ps ) -> -. A. x e. A ( ph -> ps ) )
15-Jan-2026	df-uspgren 15994	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 e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } }
11-Jan-2026	en2prde 7389	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 -> E. a E. b ( a =/= b /\ V = { a , b } ) )
10-Jan-2026	pw1mapen 16533	Equinumerosity of ( ~P 1o ^m A ) and the set of subsets of A. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- ( A e. V -> ( ~P 1o ^m A ) ~~ ~P A )
10-Jan-2026	pw1if 7433	Expressing a truth value in terms of an if expression. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- ( A e. ~P 1o -> if ( A = 1o , 1o , (/) ) = A )
10-Jan-2026	pw1m 7432	A truth value which is inhabited is equal to true. This is a variation of pwntru 4287 and pwtrufal 16534. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- ( ( A e. ~P 1o /\ E. x x e. A ) -> A = 1o )
10-Jan-2026	1ndom2 7046	Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- -. 2o ~<_ 1o
9-Jan-2026	pw1map 16532	Mapping between ( ~P 1o ^m A ) and subsets of A. (Contributed by Jim Kingdon, 9-Jan-2026.)
|- F = ( s e. ( ~P 1o ^m A ) |-> { z e. A | ( s ` z ) = 1o } )   =>    |- ( A e. V -> F : ( ~P 1o ^m A ) -1-1-onto-> ~P A )
9-Jan-2026	iftrueb01 7431	Using an if expression to represent a truth value by (/) or 1o. Unlike some theorems using if, ph does not need to be decidable. (Contributed by Jim Kingdon, 9-Jan-2026.)
|- ( if ( ph , 1o , (/) ) = 1o <-> ph )
8-Jan-2026	pfxclz 11250	Closure of the prefix extractor. This extends pfxclg 11249 from NN0 to ZZ (negative lengths are trivial, resulting in the empty word). (Contributed by Jim Kingdon, 8-Jan-2026.)
|- ( ( S e. Word A /\ L e. ZZ ) -> ( S prefix L ) e. Word A )
8-Jan-2026	fnpfx 11248	The domain of the prefix extractor. (Contributed by Jim Kingdon, 8-Jan-2026.)
|- prefix Fn ( _V X. NN0 )
7-Jan-2026	pr1or2 7390	An unordered pair, with decidable equality for the specified elements, has either one or two elements. (Contributed by Jim Kingdon, 7-Jan-2026.)
|- ( ( A e. C /\ B e. D /\ DECID A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
6-Jan-2026	upgr1elem1 15961	Lemma for upgr1edc 15962. (Contributed by AV, 16-Oct-2020.) (Revised by Jim Kingdon, 6-Jan-2026.)
|- ( ph -> { B , C } e. S )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> DECID B = C )   =>    |- ( ph -> { { B , C } } C_ { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )
3-Jan-2026	df-umgren 15935	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 e. ~P v | x ~~ 2o } }
3-Jan-2026	df-upgren 15934	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 15935). (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 e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } }
3-Jan-2026	dom1o 6997	Two ways of saying that a set is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A e. V -> ( 1o ~<_ A <-> E. j j e. A ) )
3-Jan-2026	en2m 6994	A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 2o -> E. x x e. A )
3-Jan-2026	en1m 6974	A set with one element is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 1o -> E. x x e. A )
31-Dec-2025	pw0ss 15924	There are no inhabited subsets of the empty set. (Contributed by Jim Kingdon, 31-Dec-2025.)
|- { s e. ~P (/) | E. j j e. s } = (/)
31-Dec-2025	df-ushgrm 15911	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 e. ~P v | E. j j e. s } }
29-Dec-2025	df-uhgrm 15910	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 e. ~P v | E. j j e. s } }
29-Dec-2025	iedgex 15860	Applying the indexed edge function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
|- ( G e. V -> (iEdg ` G ) e. _V )
29-Dec-2025	vtxex 15859	Applying the vertex function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
|- ( G e. V -> (Vtx ` G ) e. _V )
29-Dec-2025	snmb 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 e. _V <-> E. x x e. { A } )
27-Dec-2025	lswex 11155	Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11152 or lswcl 11154 if you want more specific results for empty or nonempty words, respectively. (Contributed by Jim Kingdon, 27-Dec-2025.)
|- ( W e. Word V -> (lastS ` W ) e. _V )
23-Dec-2025	fzowrddc 11218	Decidability of whether a range of integers is a subset of a word's domain. (Contributed by Jim Kingdon, 23-Dec-2025.)
|- ( ( S e. Word A /\ F e. ZZ /\ L e. ZZ ) -> DECID ( F..^ L ) C_ dom S )
19-Dec-2025	ccatclab 11161	The concatenation of words over two sets is a word over the union of those sets. (Contributed by Jim Kingdon, 19-Dec-2025.)
|- ( ( S e. Word A /\ T e. Word B ) -> ( S ++ T ) e. Word ( A u. B ) )
18-Dec-2025	lswwrd 11150	Extract the last symbol of a word. (Contributed by Alexander van der Vekens, 18-Mar-2018.) (Revised by Jim Kingdon, 18-Dec-2025.)
|- ( W e. Word V -> (lastS ` W ) = ( W ` ( (♯ ` W ) - 1 ) ) )
14-Dec-2025	2strstrndx 13191	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 e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> G Struct <. ( Base ` ndx ) , N >. )
12-Dec-2025	funiedgdm2vald 15873	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 e. _V   &    |- B e. _V   &    |- ( ph -> G e. X )   &    |- ( ph -> Fun ( G \ { (/) } ) )   &    |- ( ph -> A =/= B )   &    |- ( ph -> { A , B } C_ dom G )   =>    |- ( ph -> (iEdg ` G ) = (.ef ` G ) )
11-Dec-2025	funvtxdm2vald 15872	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 e. _V   &    |- B e. _V   &    |- ( ph -> G e. X )   &    |- ( ph -> Fun ( G \ { (/) } ) )   &    |- ( ph -> A =/= B )   &    |- ( ph -> { A , B } C_ dom G )   =>    |- ( ph -> (Vtx ` G ) = ( Base ` G ) )
11-Dec-2025	funiedgdm2domval 15871	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 e. V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (iEdg ` G ) = (.ef ` G ) )
11-Dec-2025	funvtxdm2domval 15870	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 e. V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (Vtx ` G ) = ( Base ` G ) )
4-Dec-2025	hash2en 11097	Two equivalent ways to say a set has two elements. (Contributed by Jim Kingdon, 4-Dec-2025.)
|- ( V ~~ 2o <-> ( V e. Fin /\ (♯ ` V ) = 2 ) )
30-Nov-2025	nninfnfiinf 16561	An element of ℕ∞ which is not finite is infinite. (Contributed by Jim Kingdon, 30-Nov-2025.)
|- ( ( A e. ℕ∞ /\ -. E. n e. om A = ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) -> A = ( i e. om |-> 1o ) )
30-Nov-2025	eluz3nn 9791	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 e. ( ZZ>= ` 3 ) -> N e. NN )
27-Nov-2025	psrelbasfi 14680	Simpler form of psrelbas 14679 when the index set is finite. (Contributed by Jim Kingdon, 27-Nov-2025.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- ( ph -> I e. Fin )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : ( NN0 ^m I ) --> K )
26-Nov-2025	mplsubgfileminv 14704	Lemma for mplsubgfi 14705. 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 )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- N = ( invg ` S )   =>    |- ( ph -> ( N ` X ) e. U )
26-Nov-2025	mplsubgfilemcl 14703	Lemma for mplsubgfi 14705. 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 )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- .+ = ( +g ` S )   =>    |- ( ph -> ( X .+ Y ) e. U )
25-Nov-2025	nninfinfwlpo 7370	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 7323). (Contributed by Jim Kingdon, 25-Nov-2025.)
|- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) <-> om e. WOmni )
23-Nov-2025	psrbagfi 14677	A finite index set gives a simpler expression for finite bags. (Contributed by Jim Kingdon, 23-Nov-2025.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( I e. Fin -> D = ( NN0 ^m I ) )
22-Nov-2025	df-acnm 7375	Define a local and length-limited version of the axiom of choice. The definition of the predicate X e. AC A is that for all families of inhabited subsets of X indexed on A (i.e. functions A --> { z e. ~P X | E. j j e. 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 e. _V /\ A. f e. ( { z e. ~P x | E. j j e. z } ^m A ) E. g A. y e. A ( g ` y ) e. ( f ` y ) ) }
21-Nov-2025	mplsubgfilemm 14702	Lemma for mplsubgfi 14705. There exists a polynomial. (Contributed by Jim Kingdon, 21-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> E. j j e. U )
15-Nov-2025	uzuzle35 9789	An integer greater than or equal to 5 is an integer greater than or equal to 3. (Contributed by AV, 15-Nov-2025.)
|- ( A e. ( ZZ>= ` 5 ) -> A e. ( ZZ>= ` 3 ) )
14-Nov-2025	2omapen 16531	Equinumerosity of ( 2o ^m A ) and the set of decidable subsets of A. (Contributed by Jim Kingdon, 14-Nov-2025.)
|- ( A e. V -> ( 2o ^m A ) ~~ { x e. ~P A | A. y e. A DECID y e. x } )
12-Nov-2025	2omap 16530	Mapping between ( 2o ^m A ) and decidable subsets of A. (Contributed by Jim Kingdon, 12-Nov-2025.)
|- F = ( s e. ( 2o ^m A ) |-> { z e. A | ( s ` z ) = 1o } )   =>    |- ( A e. V -> F : ( 2o ^m A ) -1-1-onto-> { x e. ~P A | A. y e. A DECID y e. x } )
11-Nov-2025	domomsubct 16538	A set dominated by om is subcountable. (Contributed by Jim Kingdon, 11-Nov-2025.)
|- ( A ~<_ om -> E. s ( s C_ om /\ E. f f : s -onto-> A ) )
10-Nov-2025	prdsbaslemss 13347	Lemma for prdsbas 13349 and similar theorems. (Contributed by Jim Kingdon, 10-Nov-2025.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   &    |- A = ( E ` P )   &    |- E = Slot ( E ` ndx )   &    |- ( E ` ndx ) e. NN   &    |- ( ph -> T e. X )   &    |- ( ph -> { <. ( E ` ndx ) , T >. } C_ P )   =>    |- ( ph -> A = T )
5-Nov-2025	fnmpl 14697	mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
|- mPoly Fn ( _V X. _V )
4-Nov-2025	mplelbascoe 14696	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 e. V /\ R e. W ) -> ( X e. U <-> ( X e. B /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = .0. ) ) ) )
4-Nov-2025	mplbascoe 14695	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 e. V /\ R e. W ) -> U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) } )
4-Nov-2025	mplvalcoe 14694	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 e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) }   =>    |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )
1-Nov-2025	ficardon 7384	The cardinal number of a finite set is an ordinal. (Contributed by Jim Kingdon, 1-Nov-2025.)
|- ( A e. Fin -> ( card ` A ) e. On )
31-Oct-2025	bitsdc 12498	Whether a bit is set is decidable. (Contributed by Jim Kingdon, 31-Oct-2025.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> DECID M e. (bits ` N ) )
28-Oct-2025	nn0maxcl 11776	The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> sup ( { A , B } , RR , < ) e. NN0 )
28-Oct-2025	qdcle 10496	Rational <_ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A <_ B )
17-Oct-2025	plycoeid3 15471	Reconstruct a polynomial as an explicit sum of the coefficient function up to an index no smaller than the degree of the polynomial. (Contributed by Jim Kingdon, 17-Oct-2025.)
|- ( ph -> D e. NN0 )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> ( A " ( ZZ>= ` ( D + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> M e. ( ZZ>= ` D ) )   &    |- ( ph -> X e. CC )   =>    |- ( ph -> ( F ` X ) = sum_ j e. ( 0 ... M ) ( ( A ` j ) x. ( X ^ j ) ) )
13-Oct-2025	tpfidceq 7115	A triple is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> A e. D )   &    |- ( ph -> B e. D )   &    |- ( ph -> C e. D )   &    |- ( ph -> A. x e. D A. y e. D DECID x = y )   =>    |- ( ph -> { A , B , C } e. Fin )
13-Oct-2025	prfidceq 7113	A pair is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   &    |- ( ph -> A. x e. C A. y e. C DECID x = y )   =>    |- ( ph -> { A , B } e. Fin )
13-Oct-2025	dcun 3602	The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) (Revised by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> DECID C e. A )   &    |- ( ph -> DECID C e. B )   =>    |- ( ph -> DECID C e. ( A u. B ) )
9-Oct-2025	dvdsfi 12801	A natural number has finitely many divisors. (Contributed by Jim Kingdon, 9-Oct-2025.)
|- ( N e. NN -> { x e. NN | x || N } e. Fin )
7-Oct-2025	df-mplcoe 14668	Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). The index set (which has an element for each variable) is i, the coefficients are in ring r, and for each variable there is a "degree" such that the coefficient is zero for a term where the powers are all greater than those degrees. (Degree is in quotes because there is no guarantee that coefficients below that degree are nonzero, as we do not assume decidable equality for r). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 7-Oct-2025.)
|- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / w ]_ ( w ↾s { f e. ( Base ` w ) | E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) } ) )
6-Oct-2025	dvconstss 15412	Derivative of a constant function defined on an open set. (Contributed by Jim Kingdon, 6-Oct-2025.)
|- ( ph -> S e. { RR , CC } )   &    |- J = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> X e. J )   &    |- ( ph -> A e. CC )   =>    |- ( ph -> ( S _D ( X X. { A } ) ) = ( X X. { 0 } ) )