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 1-Jul-2026 at 7:13 AM ET.

Recent Additions to the Intuitionistic Logic Explorer

Date	Label	Description
Theorem
19-Jun-2026	ringen1zr0 14560	The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rngen1zr0 14201, and semirings, see srgen1zr0 14231. (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   &    |- Z = ( 0g ` R )   =>    |- ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
19-Jun-2026	srg1zr 14230	The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   =>    |- ( ( ( R e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
18-Jun-2026	rngen1zr0 14201	The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 18-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. .0. , .0. >. , .0. >. } /\ .* = { <. <. .0. , .0. >. , .0. >. } ) ) )
18-Jun-2026	rngen1zr 14200	The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 18-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   =>    |- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
18-Jun-2026	rng1zr 14199	The only ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   =>    |- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
18-Jun-2026	rng1zrlem 14198	Lemma for rng1zr 14199 and srg1zr 14230. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   =>    |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
17-Jun-2026	ballotfi 13226	Bertrand's ballot problem : the probability that A is ahead throughout the counting. The proof formalized here is a proof "by reflection", as opposed to other known proofs "by induction" or "by permutation". This is Metamath 100 proof #30. (Contributed by Thierry Arnoux, 7-Dec-2016.) (Revised by Jim Kingdon, 17-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   =>    |- ( P ` E ) = ( ( M - N ) / ( M + N ) )
17-Jun-2026	ballotfilembfi 13183	The set of countings where B got the first vote is finite. (Contributed by Jim Kingdon, 17-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   =>    |- { c e. ( O \ E ) | -. 1 e. c } e. Fin
17-Jun-2026	ballotfilemafi 13182	The set of countings where A got the first vote, but does not stay strictly ahead throughout, is finite. (Contributed by Jim Kingdon, 17-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   =>    |- { c e. ( O \ E ) | 1 e. c } e. Fin
17-Jun-2026	ballotfilemefi 13181	E is finite. (Contributed by Jim Kingdon, 17-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   =>    |- E e. Fin
17-Jun-2026	rabxmdc 3544	Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Jim Kingdon, 17-Jun-2026.)
|- ( A. x e. A DECID ph -> A = ( { x e. A | ph } u. { x e. A | -. ph } ) )
15-Jun-2026	ballotfilemgun 13212	A property of the defined .^ operator. (Contributed by Thierry Arnoux, 26-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   &    |- ( ph -> U e. O )   &    |- ( ph -> L e. ( J ... K ) )   =>    |- ( ph -> ( U .^ ( J ... K ) ) = ( ( U .^ ( J ... ( L - 1 ) ) ) + ( U .^ ( L ... K ) ) ) )
15-Jun-2026	ballotfilemgval 13211	Expand the value of .^. (Contributed by Thierry Arnoux, 21-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   &    |- ( ph -> U e. O )   &    |- ( ph -> J e. ZZ )   &    |- ( ph -> K e. ZZ )   &    |- ( ph -> V = ( J ... K ) )   =>    |- ( ph -> ( U .^ V ) = ( (♯ ` ( V i^i U ) ) - (♯ ` ( V \ U ) ) ) )
15-Jun-2026	ballotfilemdifcfz 13171	Lemma for ballotfi . The portion of a counting representing votes for B within a specified integer range is finite. (Contributed by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> J e. ZZ )   &    |- ( ph -> K e. ZZ )   =>    |- ( ph -> ( ( J ... K ) \ C ) e. Fin )
15-Jun-2026	ballotfilemcinfz 13170	Lemma for ballotfi . The portion of a counting representing votes for A within a specified integer range is finite. (Contributed by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> J e. ZZ )   &    |- ( ph -> K e. ZZ )   =>    |- ( ph -> ( ( J ... K ) i^i C ) e. Fin )
12-Jun-2026	ballotfilemsle 13192	The infimum of the set of zeroes of F is a lower bound. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 }   =>    |- ( ( C e. ( O \ E ) /\ X e. S ) -> inf ( S , RR , < ) <_ X )
12-Jun-2026	ballotfilemscl 13191	The set of zeroes of F has an infimum. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 }   =>    |- ( C e. ( O \ E ) -> inf ( S , RR , < ) e. S )
12-Jun-2026	infssfzledc 10619	The infimum of a decidable inhabited subset of an integer range is a lower bound for that set. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- S = { n e. ( M ... N ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) <_ A )
12-Jun-2026	infssfzcldc 10618	The infimum of a decidable inhabited subset of an integer range is a member of the set. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- S = { n e. ( M ... N ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) e. S )
8-Jun-2026	ballotfilemdifcfi 13169	Lemma for ballotfi . The portion of a counting representing votes for B up to a specified integer is finite. (Contributed by Jim Kingdon, 8-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> J e. ZZ )   =>    |- ( ph -> ( ( 1 ... J ) \ C ) e. Fin )
8-Jun-2026	ballotfilemcinfi 13168	Lemma for ballotfi . The portion of a counting representing votes for A up to a specified integer is finite. (Contributed by Jim Kingdon, 8-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> J e. ZZ )   =>    |- ( ph -> ( ( 1 ... J ) i^i C ) e. Fin )
8-Jun-2026	zfidc 9673	Whether an integer is an element of a finite set of integers is decidable. (Contributed by Jim Kingdon, 8-Jun-2026.)
|- ( ( S C_ ZZ /\ A e. ZZ /\ S e. Fin ) -> DECID A e. S )
7-Jun-2026	ballotfilemcdc 13167	Lemma for ballotfi . It is decidable whether a given integer is an element of a particular element of O. (Contributed by Jim Kingdon, 7-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> K e. ZZ )   =>    |- ( ph -> DECID K e. C )
5-Jun-2026	hashpwfi 11218	The number of finite subsets of a finite set is two raised to the power of the size of the set. For a similar theorem with set size expressed using equinumerosity, see 2omapfi 7284. For the number of subsets (which need not be finite) of a set, see pw1mapen 16896. (Contributed by Jim Kingdon, 5-Jun-2026.)
|- ( A e. Fin -> (♯ ` ( ~P A i^i Fin ) ) = ( 2 ^ (♯ ` A ) ) )
4-Jun-2026	ballotfilemonn 13165	The size of the universe is at least one. (Contributed by Jim Kingdon, 4-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   =>    |- (♯ ` O ) e. NN
3-Jun-2026	papeq2 7574	Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
|- ( A = B -> ( R Ap A <-> R Ap B ) )
3-Jun-2026	papeq1 7573	Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
|- ( R = S -> ( R Ap A <-> S Ap A ) )
2-Jun-2026	resq01 11044	If a real number equals its square, it must be 0 or 1. (Contributed by Jim Kingdon, 2-Jun-2026.)
|- ( A e. RR -> ( ( A ^ 2 ) = A <-> ( A = 0 \/ A = 1 ) ) )
31-May-2026	aprprop 14539	If two structures have the same ring components (properties), df-apr 14528 generates the same relation for both of them. (Contributed by Jim Kingdon, 31-May-2026.)
|- ( Base ` K ) = ( Base ` L )   &    |- ( +g ` K ) = ( +g ` L )   &    |- ( .r ` K ) = ( .r ` L )   =>    |- ( K e. Ring -> (#r ` K ) = (#r ` L ) )
31-May-2026	ringunitsap0 14532	The set of units of a ring. If R is a local ring, # is an apartness and this theorem states that the units of a ring are those elements apart from zero (see aprlring 14538). Given the definition of #r this theorem holds even if # is not an apartness, however. (Contributed by Jim Kingdon, 31-May-2026.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- # = (#r ` R )   =>    |- ( R e. Ring -> { x e. B | x # .0. } = (Unit ` R ) )
30-May-2026	ringunitap 14531	Elementhood in the set of units. (Contributed by Jim Kingdon, 30-May-2026.)
|- B = ( Base ` R )   &    |- U = (Unit ` R )   &    |- .0. = ( 0g ` R )   &    |- # = (#r ` R )   =>    |- ( R e. Ring -> ( X e. U <-> ( X e. B /\ X # .0. ) ) )
29-May-2026	drnglring 14545	A division ring is a local ring. (Contributed by Jim Kingdon, 29-May-2026.)
|- ( R e. DivRing -> R e. LRing )
29-May-2026	isdrngtap 14544	The predicate "is a division ring". (Contributed by Jim Kingdon, 29-May-2026.)
|- B = ( Base ` R )   &    |- # = (#r ` R )   =>    |- ( R e. DivRing <-> ( R e. Ring /\ # TAp B ) )
29-May-2026	df-drngap 14542	Define class of all division rings. A division ring is a ring in which the relation given by df-apr 14528 is a tight apartness. (Contributed by Jim Kingdon, 29-May-2026.)
|- DivRing = { r e. Ring | (#r ` r ) TAp ( Base ` r ) }
29-May-2026	aprunit 14530	The df-apr 14528 relation with zero expresses whether a ring element is a unit. That is, the difference of an element of a ring and zero is invertible iff the element is a unit. (Contributed by Jim Kingdon, 29-May-2026.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- U = (Unit ` R )   &    |- # = (#r ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X # .0. <-> X e. U ) )
29-May-2026	tapap 7580	A tight apartness is an apartness. (Contributed by Jim Kingdon, 29-May-2026.)
|- ( R TAp A -> R Ap A )
28-May-2026	aprlring 14538	A ring is a local ring if and only if the relation given by df-apr 14528 is an apartness relation. (Contributed by Jim Kingdon, 28-May-2026.)
|- ( R e. Ring -> ( R e. LRing <-> (#r ` R ) Ap ( Base ` R ) ) )
28-May-2026	papcotr 7577	An apartness is cotransitive. (Contributed by Jim Kingdon, 28-May-2026.)
|- ( ph -> R Ap A )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. A )   &    |- ( ph -> X R Y )   &    |- ( ph -> Z e. A )   =>    |- ( ph -> ( X R Z \/ Y R Z ) )
27-May-2026	trimul0or 16971	Real number trichotomy implies that if a product is zero, one of its factors must be zero. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) -> A. u e. CC A. v e. CC ( ( u x. v ) = 0 -> ( u = 0 \/ v = 0 ) ) )
27-May-2026	aprnzr 14537	If the relation given by df-apr 14528 on a ring is an apartness relation, then the ring is a nonzero ring. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) -> R e. NzRing )
27-May-2026	papsym 7576	An apartness is symmetric. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( ph -> R Ap A )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. A )   &    |- ( ph -> X R Y )   =>    |- ( ph -> Y R X )
27-May-2026	papirr 7575	An apartness is irreflexive. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( ( R Ap A /\ X e. A ) -> -. X R X )
24-May-2026	gfsumz 14109	Value of a finite group sum over the zero element. (Contributed by Jim Kingdon, 24-May-2026.)
|- .0. = ( 0g ` G )   =>    |- ( ( G e. CMnd /\ A e. Fin ) -> ( G gfsumgf ( k e. A |-> .0. ) ) = .0. )
22-May-2026	sshashneg 11230	Subsets of a class of a negative size (a degenerate case). Together with ssenneg 11229 this shows that sseqn 11228 could not be extended beyond N e. NN0. (Contributed by Jim Kingdon, 22-May-2026.)
|- ( ( N e. ZZ /\ N < 0 ) -> { x e. ( ~P A i^i Fin ) | (♯ ` x ) = N } = (/) )
22-May-2026	ssenneg 11229	Subsets of a class of a negative size (a degenerate case). Together with sshashneg 11230 this shows that sseqn 11228 could not be extended beyond N e. NN0. (Contributed by Jim Kingdon, 22-May-2026.)
|- ( ( N e. ZZ /\ N < 0 ) -> { x e. ~P A | x ~~ ( 1 ... N ) } = { (/) } )
22-May-2026	sseqn 11228	Two ways to express the subsets of a class of a given size. It might seem that { x e. ~P A | (♯ ` x ) = N } would suffice, but that would require the converse of hashcl 11169 or something similar. Although each side of the equality would be well defined if we changed N e. NN0 to N e. ZZ, they would give different results for the (degenerate) case of a negative size, as shown at ssenneg 11229 and sshashneg 11230. (Contributed by Jim Kingdon, 22-May-2026.)
|- ( N e. NN0 -> { x e. ~P A | x ~~ ( 1 ... N ) } = { x e. ( ~P A i^i Fin ) | (♯ ` x ) = N } )
20-May-2026	ballotfilemofi 13163	O is finite. (Contributed by Jim Kingdon, 20-May-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   =>    |- O e. Fin
19-May-2026	fipwfi 7285	The set of finite subsets of a finite set is finite. (Contributed by Jim Kingdon, 19-May-2026.)
|- ( A e. Fin -> ( ~P A i^i Fin ) e. Fin )
18-May-2026	2omapfi 7284	The number of finite subsets of a finite set. For a similar theorem with set size expressed using ♯ (df-ihash 11164), see hashpwfi 11218. (Contributed by Jim Kingdon, 18-May-2026.)
|- ( A e. Fin -> ( 2o ^m A ) ~~ ( ~P A i^i Fin ) )
18-May-2026	fissfi 7229	A finite subset of a finite set is a decidable subset. (Contributed by Jim Kingdon, 18-May-2026.)
|- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> A. x e. A DECID x e. S )
18-May-2026	fresaunres1disj 5551	From the union of two functions with disjoint domains, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.) (Revised by Jim Kingdon, 18-May-2026.)
|- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` A ) = F )
18-May-2026	fresaunres2disj 5550	From the union of two functions with disjoint domains, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Jim Kingdon, 18-May-2026.)
|- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = G )
15-May-2026	fsuppcorn 7267	The composition of a 1-1 function with a finitely supported function is finitely supported. The purpose of the ( F supp Z ) C_ ran G condition is to ensure we don't subset the support of the function in such a way as to fun afoul of exmidssfi 7212. (Other alternative conditions might also be sufficient). (Contributed by AV, 28-May-2019.) (Revised by Jim Kingdon, 15-May-2026.)
|- ( ph -> F finSupp Z )   &    |- ( ph -> G : X -1-1-> Y )   &    |- ( ph -> Z e. W )   &    |- ( ph -> F e. V )   &    |- ( ph -> G e. U )   &    |- ( ph -> ( F supp Z ) C_ ran G )   =>    |- ( ph -> ( F o. G ) finSupp Z )
13-May-2026	lincmble 10356	A linear combination of two reals which lies in the interval between them. Like lincmb01cmp 10355 but generalized to require merely A <_ B not A < B. (Contributed by Jim Kingdon, 13-May-2026.)
|- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) )
5-May-2026	fmelpw1o 7570	With a formula ph one can associate an element of ~P 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than T. and F., by nndc 859, which translate to 1o and (/) respectively by iftrue 3631 and iffalse 3634, giving pwtrufal 16897). As proved in if0ab 3627, the associated element of ~P 1o is the extension, in ~P 1o, of the formula ph. (Contributed by BJ, 15-Aug-2024.) (Proof shortened by BJ, 5-May-2026.)
|- if ( ph , 1o , (/) ) e. ~P 1o
5-May-2026	if0elpw 4276	A conditional class with the False alternative being sent to the empty class is an element of the powerset of the class corresponding to the True alternative when that class is a set. This statement requires fewer axioms than the general case ifelpwung 4607. (Contributed by BJ, 5-May-2026.)
|- ( A e. V -> if ( ph , A , (/) ) e. ~P A )
5-May-2026	if0ss 3628	A conditional class with the False alternative being sent to the empty class is included in the class corresponding to the True alternative. (Contributed by BJ, 5-May-2026.)
|- if ( ph , A , (/) ) C_ A
27-Apr-2026	repiecef 16938	Piecewise definition on the reals yields a function. The function agrees with F and G on their respective parts of the real line; see repiecele0 16936 and repiecege0 16937. From an online post by James E Hanson. The construction was published in Martín Hötzel Escardó, "Effective and sequential definition by cases on the reals via infinite signed-digit numerals", Electronic Notes in Theoretical Computer Science 10 (1998), page 2, https://martinescardo.github.io/papers/lexnew.pdf. 16937 (Contributed by Jim Kingdon, 27-Apr-2026.)
|- ( ph -> F : ( -oo (,] 0 ) --> RR )   &    |- ( ph -> G : ( 0 [,) +oo ) --> RR )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   &    |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )   =>    |- ( ph -> H : RR --> RR )
27-Apr-2026	repiecege0 16937	Piecewise definition on the reals agrees with the nonnegative part of the definition. See repiecef 16938 for more on this construction. (Contributed by Jim Kingdon, 27-Apr-2026.)
|- ( ph -> F : ( -oo (,] 0 ) --> RR )   &    |- ( ph -> G : ( 0 [,) +oo ) --> RR )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   &    |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )   =>    |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( H ` A ) = ( G ` A ) )
27-Apr-2026	repiecele0 16936	Piecewise definition on the reals agrees with the nonpositive part of the definition. See repiecef 16938 for more on this construction. (Contributed by Jim Kingdon, 27-Apr-2026.)
|- ( ph -> F : ( -oo (,] 0 ) --> RR )   &    |- ( ph -> G : ( 0 [,) +oo ) --> RR )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   &    |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )   =>    |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( H ` A ) = ( F ` A ) )
27-Apr-2026	repiecelem 16935	Lemma for repiecele0 16936, repiecege0 16937, and repiecef 16938. The function H is defined everywhere. (Contributed by Jim Kingdon, 27-Apr-2026.)
|- ( ph -> F : ( -oo (,] 0 ) --> RR )   &    |- ( ph -> G : ( 0 [,) +oo ) --> RR )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   &    |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )   =>    |- ( ( ph /\ A e. RR ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
24-Apr-2026	qdiff 16959	The rationals are exactly those reals for which there exist two distinct rationals that are the same distance from the original number. Similar to apdiff 16958 but by stating the result positively we can completely sidestep the issue of not equal versus apart in the statement of the result. From an online post by Ingo Blechschmidt. (Contributed by Jim Kingdon, 24-Apr-2026.)
|- ( A e. RR -> ( A e. QQ <-> E. q e. QQ E. r e. QQ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) )
23-Apr-2026	exmidpeirce 16907	Excluded middle is equivalent to Peirce's law. Read an element of ~P 1o as being a truth value and x = 1o being that x is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfrompeirce 1495. (Contributed by Jim Kingdon, 23-Apr-2026.)
|- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
22-Apr-2026	exmidcon 16906	Excluded middle is equivalent to the form of contraposition which removes negation. Read an element of ~P 1o as being a truth value and x = 1o being that x is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromcon 1494. (Contributed by Jim Kingdon, 22-Apr-2026.)
|- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) )
22-Apr-2026	exmidnotnotr 16905	Excluded middle is equivalent to double negation elimination. Read an element of ~P 1o as being a truth value and x = 1o being that x is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromnotnotr 1493. (Contributed by Jim Kingdon, 22-Apr-2026.)
|- (EXMID <-> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
18-Apr-2026	hashtpglem 11243	Lemma for hashtpg 11244. This is one of the three not-equal conclusions required for the reverse direction. (Contributed by Jim Kingdon, 18-Apr-2026.)
|- ( ph -> A e. U )   &    |- ( ph -> B e. V )   &    |- ( ph -> C e. W )   &    |- ( ph -> (♯ ` { A , B , C } ) = 3 )   =>    |- ( ph -> B =/= C )
17-Apr-2026	hashtpgim 11242	The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.) (Revised by Jim Kingdon, 17-Apr-2026.)
|- ( ( A e. U /\ B e. V /\ C e. W ) -> ( ( A =/= B /\ B =/= C /\ C =/= A ) -> (♯ ` { A , B , C } ) = 3 ) )
14-Apr-2026	depind 16630	Theorem related to a dependently typed induction principle in type theory. (Contributed by Matthew House, 14-Apr-2026.)
|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   =>    |- ( ph -> E! f e. X_ n e. NN0 ( P ` n ) ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) )
14-Apr-2026	depindlem3 16629	Lemma for depind 16630. (Contributed by Matthew House, 14-Apr-2026.)
|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> A. f e. X_ n e. NN0 ( P ` n ) ( ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) -> f = F ) )
14-Apr-2026	depindlem2 16628	Lemma for depind 16630. (Contributed by Matthew House, 14-Apr-2026.)
|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> F e. X_ n e. NN0 ( P ` n ) )
14-Apr-2026	depindlem1 16627	Lemma for depind 16630. (Contributed by Matthew House, 14-Apr-2026.)
|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> ( F : NN0 --> _V /\ ( F ` 0 ) = A /\ A. n e. NN0 ( F ` ( n + 1 ) ) = ( ( H ` n ) ` ( F ` n ) ) ) )
8-Apr-2026	gfsumcl 14110	Closure of a finite group sum. (Contributed by Jim Kingdon, 8-Apr-2026.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> ( G gfsumgf F ) e. B )
4-Apr-2026	gsumsplit0 14099	Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gsumsplit1r 13661 except that N can equal M - 1. (Contributed by Jim Kingdon, 4-Apr-2026.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ( ZZ>= ` ( M - 1 ) ) )   &    |- ( ph -> F : ( M ... ( N + 1 ) ) --> B )   =>    |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
4-Apr-2026	fzf1o 12086	A finite set can be enumerated by integers starting at one. (Contributed by Jim Kingdon, 4-Apr-2026.)
|- ( A e. Fin -> E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
3-Apr-2026	gfsump1 14108	Splitting off one element from a finite group sum. This would typically used in a proof by induction. (Contributed by Jim Kingdon, 3-Apr-2026.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> F : ( Y u. { Z } ) --> B )   &    |- ( ph -> Y e. Fin )   &    |- ( ph -> Z e. V )   &    |- ( ph -> -. Z e. Y )   =>    |- ( ph -> ( G gfsumgf F ) = ( ( G gfsumgf ( F |` Y ) ) .+ ( F ` Z ) ) )
2-Apr-2026	gfsumsn 14107	Group sum of a singleton. (Contributed by Jim Kingdon, 2-Apr-2026.)
|- B = ( Base ` G )   &    |- ( k = M -> A = C )   =>    |- ( ( G e. CMnd /\ M e. V /\ C e. B ) -> ( G gfsumgf ( k e. { M } |-> A ) ) = C )
31-Mar-2026	sspw1or2 7508	The set of subsets of a given set with one or two elements can be expressed as elements of the power set or as inhabited elements of the power set. (Contributed by Jim Kingdon, 31-Mar-2026.)
|- { x e. { s e. ~P V | E. j j e. s } | ( x ~~ 1o \/ x ~~ 2o ) } = { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) }
28-Mar-2026	imaf1fi 7206	The image of a finite set under a one-to-one mapping is finite. (Contributed by Jim Kingdon, 28-Mar-2026.)
|- ( ( F : A -1-1-> B /\ X C_ A /\ X e. Fin ) -> ( F " X ) e. Fin )
26-Mar-2026	gsumshift 14105	Shifting the indexes of a group sum indexed by consecutive integers. (Contributed by Jim Kingdon, 26-Mar-2026.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) --> B )   &    |- S = ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) )   =>    |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. S ) ) )
26-Mar-2026	gfsum0 14104	An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
|- ( G e. CMnd -> ( G gfsumgf (/) ) = ( 0g ` G ) )
25-Mar-2026	gsumgfsum 14106	On an integer range, gsumg and gfsumgf agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( G gsumg F ) = ( G gfsumgf F ) )
25-Mar-2026	gsumgfsum1 14103	On an integer range starting at one, gsumg and gfsumgf agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> F : ( 1 ... N ) --> B )   =>    |- ( ph -> ( G gsumg F ) = ( G gfsumgf F ) )
24-Mar-2026	gfsumval 14102	Value of the finite group sum over an unordered finite set. (Contributed by Jim Kingdon, 24-Mar-2026.)
|- B = ( Base ` W )   &    |- ( ph -> W e. CMnd )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> G : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )   =>    |- ( ph -> ( W gfsumgf F ) = ( W gsumg ( F o. G ) ) )
23-Mar-2026	df-gfsum 14101	Define the finite group sum (iterated sum) over an unordered finite set. Given G gfsumgf F where F : A --> ( Base ` G ), the set of indices is A and the values are given by F at each index. For this notation, A is a finite set and G is a commutative monoid, and the sum adds up these elements in some order (the sum does not depend on the order). For a sum indexed by consecutive integers (and thus defining an order for the sum), see df-igsum 13556. (Contributed by Jim Kingdon, 23-Mar-2026.)
|- gfsumgf = ( w e. CMnd , f e. _V |-> ( iota x ( dom f e. Fin /\ E. g ( g : ( 1 ... (♯ ` dom f ) ) -1-1-onto-> dom f /\ x = ( w gsumg ( f o. g ) ) ) ) ) )
20-Mar-2026	exmidssfi 7212	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 <-> A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin ) )
18-Mar-2026	umgr1een 16246	A graph with one non-loop edge is a multigraph. (Contributed by Jim Kingdon, 18-Mar-2026.)
|- ( ph -> K e. X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> E ~~ 2o )   =>    |- ( ph -> <. V , { <. K , E >. } >. e. UMGraph )
18-Mar-2026	upgr1een 16245	A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16242 for a different way of specifying a graph with one edge. (Contributed by Jim Kingdon, 18-Mar-2026.)
|- ( ph -> K e. X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> E ~~ 2o )   =>    |- ( ph -> <. V , { <. K , E >. } >. e. UPGraph )
14-Mar-2026	trlsex 16508	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 16567	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 16428	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 11188	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 6447	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 16528	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 16519	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 16414	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 16411	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 16410	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 7176	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 16891	The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7183), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7179. (Contributed by Jim Kingdon, 14-Feb-2026.)
|- -. om ~<_ ~P 1o
14-Feb-2026	pw1ndom3 16890	The powerset of 1o does not dominate 3o. This is another way of saying that ~P 1o does not have three elements (like pwntru 4317). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)
|- -. 3o ~<_ ~P 1o