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 6-Nov-2026 at 7:06 AM ET.

Recent Additions to the Intuitionistic Logic Explorer

Date	Label	Description
Theorem
27-Oct-2026	psrbagmaxlem1y 15182	Lemma for psrbagmax 15183. All the bags in Y are (pointwise) less than S, and thus elements of Z. (Contributed by Jim Kingdon, 27-Oct-2026.)
|- ( ph -> X C_ ( NN0 ^m I ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> P : I --> NN0 )   &    |- ( ph -> R : I --> NN0 )   &    |- S = ( i e. I |-> sup ( { ( P ` i ) , ( R ` i ) } , RR , < ) )   &    |- Z = { q e. ( NN0 ^m I ) | q oR <_ S }   &    |- ( ph -> A. j e. I A. x e. X ( x ` j ) <_ ( P ` j ) )   &    |- ( ph -> A. j e. I A. y e. Y ( y ` j ) <_ ( R ` j ) )   &    |- ( ph -> Y C_ ( NN0 ^m I ) )   &    |- ( ph -> Y e. Fin )   =>    |- ( ph -> Y C_ Z )
27-Oct-2026	psrbagmaxlem1x 15181	Lemma for psrbagmax 15183. All the bags in X are (pointwise) less than S, and thus elements of Z. (Contributed by Jim Kingdon, 27-Oct-2026.)
|- ( ph -> X C_ ( NN0 ^m I ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> P : I --> NN0 )   &    |- ( ph -> R : I --> NN0 )   &    |- S = ( i e. I |-> sup ( { ( P ` i ) , ( R ` i ) } , RR , < ) )   &    |- Z = { q e. ( NN0 ^m I ) | q oR <_ S }   &    |- ( ph -> A. j e. I A. x e. X ( x ` j ) <_ ( P ` j ) )   &    |- ( ph -> A. j e. I A. y e. Y ( y ` j ) <_ ( R ` j ) )   &    |- ( ph -> Y C_ ( NN0 ^m I ) )   &    |- ( ph -> Y e. Fin )   =>    |- ( ph -> X C_ Z )
27-Oct-2026	psrbagmaxlemfi 15180	Lemma for psrbagmax 15183. Applying psrbaglefifi 15175 to the sets in this theorem. (Contributed by Jim Kingdon, 27-Oct-2026.)
|- ( ph -> X C_ ( NN0 ^m I ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> P : I --> NN0 )   &    |- ( ph -> R : I --> NN0 )   &    |- S = ( i e. I |-> sup ( { ( P ` i ) , ( R ` i ) } , RR , < ) )   &    |- Z = { q e. ( NN0 ^m I ) | q oR <_ S }   =>    |- ( ph -> Z e. Fin )
27-Oct-2026	psrbagmaxlem1a 15179	Lemma for psrbagmax 15183. A new bag which is the pointwise maximum of two bags. (Contributed by Jim Kingdon, 27-Oct-2026.)
|- ( ph -> X C_ ( NN0 ^m I ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> P : I --> NN0 )   &    |- ( ph -> R : I --> NN0 )   &    |- S = ( i e. I |-> sup ( { ( P ` i ) , ( R ` i ) } , RR , < ) )   =>    |- ( ph -> S : I --> NN0 )
27-Oct-2026	psrbagmaxlem1 15178	Lemma for psrbagmax 15183. Existence of a bag which is greater than any element of a finite set of bags. (Contributed by Jim Kingdon, 27-Oct-2026.)
|- ( ph -> X C_ ( NN0 ^m I ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> I e. Fin )   =>    |- ( ph -> E. p ( p : I --> NN0 /\ A. j e. I A. x e. X ( x ` j ) <_ ( p ` j ) ) )
24-Oct-2026	psrbagmax 15183	Given two finite sets of bags, provide a finite set which contains all of their elements. The proof constructs the result ( Z in the lemmas), proves its properties (via psrbagmaxlemfi 15180, psrbagmaxlem1x 15181, and psrbagmaxlem1y 15182), and then this proof combines those lemmas and assigns the result to the variable z. (Contributed by Jim Kingdon, 24-Oct-2026.)
|- ( ph -> X C_ ( NN0 ^m I ) )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> Y C_ ( NN0 ^m I ) )   &    |- ( ph -> Y e. Fin )   &    |- ( ph -> I e. Fin )   =>    |- ( ph -> E. z ( ( z C_ ( NN0 ^m I ) /\ z e. Fin ) /\ ( X C_ z /\ Y C_ z ) ) )
24-Oct-2026	fsuppfiss 7300	A finitely supported function has a finite subset of its domain such that values outside that subset map to zero. (Contributed by Jim Kingdon, 24-Oct-2026.)
|- ( ph -> F finSupp Z )   =>    |- ( ph -> E. w ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) )
22-Oct-2026	psrbagfsuppfi 15165	Finite bags have finite support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by Jim Kingdon, 22-Oct-2026.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ I e. Fin ) -> F finSupp 0 )
18-Oct-2026	suppinisegdc 6513	Relation between the support ( F supp Z ) and the initial segment ( `' F " { Z } ). (Contributed by Thierry Arnoux, 25-Jun-2024.) (Revised by Jim Kingdon, 18-Oct-2026.)
|- ( ph -> F : A --> B )   &    |- ( ph -> F e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. x e. B A. y e. B DECID x = y )   =>    |- ( ph -> ( A \ ( F supp Z ) ) = ( `' F " { Z } ) )
17-Oct-2026	funisfsuppdc 7298	The property of a function to be finitely supported (in relation to a given zero). This shows that df-fsuppfi 7293 and df-supp 6480 give the same results, in the case of decidable equality. (Contributed by AV, 23-May-2019.) (Revised by Jim Kingdon, 17-Oct-2026.)
|- ( ph -> F e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> A. x e. B A. y e. B DECID x = y )   =>    |- ( ph -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
16-Oct-2026	isfsuppfid 7297	Deduction form of isfsuppfi 7296. (Contributed by SN, 29-Jul-2024.) (Revised by Jim Kingdon, 16-Oct-2026.)
|- ( ph -> R e. V )   &    |- ( ph -> Z e. U )   &    |- ( ph -> Fun R )   &    |- ( ph -> W e. Fin )   &    |- ( ph -> W C_ dom R )   &    |- ( ph -> ( R " ( dom R \ W ) ) C_ { Z } )   =>    |- ( ph -> R finSupp Z )
16-Oct-2026	df-fsuppfi 7293	Define the property of a function to be finitely supported (in relation to a given zero). This definition will perhaps be most useful when the domain of the function has decidable equality; consider as an alternative ( R supp Z ) e. Fin (see df-supp 6480) which will behave well when a codomain of the function has decidable equality. If both domain and codomain have decidable equality, the two are equivalent as seen at funisfsuppdc 7298. (Contributed by AV, 23-May-2019.) (Revised by Jim Kingdon, 16-Oct-2026.)
|- finSupp = { <. r , z >. | ( Fun r /\ E. w ( w e. Fin /\ w C_ dom r /\ ( r " ( dom r \ w ) ) C_ { z } ) ) }
13-Oct-2026	mapm 6975	Set exponentiation is inhabited if the base is inhabited. (Contributed by Jim Kingdon, 13-Oct-2026.)
|- ( ( A e. V /\ B e. W /\ C e. A ) -> E. x x e. ( A ^m B ) )
11-Oct-2026	psrbagdceq 15167	Finite bags have decidable equality. (Contributed by Jim Kingdon, 11-Oct-2026.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( I e. Fin /\ A e. D /\ B e. D ) -> DECID A = B )
9-Oct-2026	ofovex 6311	Applying a function operation to two functions which are sets yields a set. (Contributed by Jim Kingdon, 9-Oct-2026.)
|- ( ( F e. V /\ G e. W ) -> ( F oF R G ) e. _V )
8-Oct-2026	idjufi 14268	A finite indexed disjoint union of finite sets is finite. (Contributed by Jim Kingdon, 8-Oct-2026.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ j e. A ) -> C e. Fin )   =>    |- ( ph -> U_ j e. A ( { j } X. C ) e. Fin )
8-Oct-2026	sn0disj 4127	A collection, each of whose elements is empty or a singleton of the index, is disjoint. (Contributed by Jim Kingdon, 8-Oct-2026.)
|- ( ( ph /\ x e. A ) -> ( B = (/) \/ B = { x } ) )   =>    |- ( ph -> Disj x e. A B )
6-Oct-2026	gsum2dfi 14272	Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.) (Revised by Jim Kingdon, 6-Oct-2026.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> Rel A )   &    |- ( ph -> D e. Fin )   &    |- ( ph -> dom A C_ D )   &    |- ( ph -> dom A e. Fin )   &    |- ( ph -> F : A --> B )   &    |- ( ( ph /\ j e. D ) -> ( A " { j } ) e. Fin )   =>    |- ( ph -> ( G gsumg F ) = ( G gsumg ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )
6-Oct-2026	relresfin 7270	Restricting to a subset of a finite relation's domain gives a finite set. (Contributed by Jim Kingdon, 6-Oct-2026.)
|- ( ph -> E e. Fin )   &    |- ( ph -> Rel E )   &    |- ( ph -> dom E e. Fin )   &    |- ( ph -> B C_ dom E )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> ( E |` B ) e. Fin )
4-Oct-2026	gsum2dfilem2 14271	Lemma for gsum2dfi 14272. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.) (Revised by Jim Kingdon, 4-Oct-2026.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> Rel A )   &    |- ( ph -> D e. Fin )   &    |- ( ph -> dom A C_ D )   &    |- ( ph -> dom A e. Fin )   &    |- ( ph -> F : A --> B )   &    |- ( ( ph /\ j e. D ) -> ( A " { j } ) e. Fin )   =>    |- ( ph -> ( G gsumg F ) = ( G gsumg ( j e. dom A |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )
2-Oct-2026	idjudju 17269	An indexed disjoint union over two sets equals the disjoint union of those two sets. This connects our notation for an indexed disjoint union, U_ x e. C ( { x } X. F ) which uses df-iun 4014, with our notation for the disjoint union of two sets (df-dju 7384). (Contributed by Jim Kingdon, 2-Oct-2026.)
|- F = if ( x = (/) , A , B )   =>    |- U_ x e. 2o ( { x } X. F ) = ( A ⊔ B )
2-Oct-2026	gsum2dfilem1 14270	Lemma 1 for gsum2dfi 14272. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.) (Revised by Jim Kingdon, 2-Oct-2026.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> F : A --> B )   &    |- ( ( ph /\ j e. D ) -> ( A " { j } ) e. Fin )   =>    |- ( ( ph /\ j e. D ) -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )
2-Oct-2026	idjusnfi 14269	The image of a singleton through an indexed disjoint union, where it is of finite sets. (Contributed by Jim Kingdon, 2-Oct-2026.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ j e. A ) -> C e. Fin )   &    |- U = U_ j e. A ( { j } X. C )   =>    |- ( ( ph /\ K e. A ) -> ( U " { K } ) e. Fin )
24-Sep-2026	gsumresfi 14262	Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 24-Sep-2026.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> A. x e. ( A \ W ) ( F ` x ) = .0. )   &    |- ( ph -> W e. Fin )   &    |- ( ph -> W C_ A )   =>    |- ( ph -> ( G gsumg ( F |` W ) ) = ( G gsumg F ) )
23-Sep-2026	sineq0re 16093	A complex number whose sine is zero is real. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.) (Revised by Jim Kingdon, 23-Sep-2026.)
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> A e. RR )
18-Sep-2026	rirrdisj 17308	The rational and irrational numbers are disjoint. Here irrational means apart from any rational number. (Contributed by Jim Kingdon, 18-Sep-2026.)
|- ( QQ i^i { x e. RR | A. q e. QQ x # q } ) = (/)
18-Sep-2026	efnthr 16193	An equation involving an N-th power. The expression exp ( 2 pi _i / N ) is a way to write the primitive N-th root of unity with the smallest positive argument. We could write it as -u 1 ^c ( 2 / N ) except that we only support positive real numbers as the base for ^c. The converse is presumably also provable but we do not have a proof yet. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by Jim Kingdon, 18-Sep-2026.)
|- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> ( E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) -> ( A ^ N ) = B ) )
16-Sep-2026	cntzm 14160	If the centralizer of a subset of a magma has an element, the magma is inhabited. (Contributed by Jim Kingdon, 16-Sep-2026.)
|- Z = (Cntz ` M )   =>    |- ( X e. ( Z ` S ) -> E. w w e. M )
16-Sep-2026	ressmex 13477	If a structure restriction is inhabited, the structure is a set and so is the class it is restricted to. (Contributed by Jim Kingdon, 16-Sep-2026.)
|- R = ( W ↾s A )   =>    |- ( X e. R -> ( W e. _V /\ A e. _V ) )
15-Sep-2026	cntzex 14149	Set existence of the centralizer. (Contributed by Jim Kingdon, 15-Sep-2026.)
|- Z = (Cntz ` M )   =>    |- ( M e. V -> Z e. _V )
9-Sep-2026	flaplt 10738	The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005.) (Revised by Jim Kingdon, 9-Sep-2026.)
|- ( ( ( A e. QQ \/ ( A e. RR /\ A. q e. QQ A # q ) ) /\ B e. ZZ ) -> ( A < B <-> ( |_ ` A ) < B ) )
8-Sep-2026	fiidxsupcl 12017	A set of integers indexed by a finite set has an upper bound. (Contributed by Jim Kingdon, 8-Sep-2026.)
|- ( ph -> F : A --> ZZ )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> E. x e. ZZ A. y e. A ( F ` y ) <_ x )
6-Sep-2026	ofrfidc 7323	Decidability of a relation applied to two functions. (Contributed by Jim Kingdon, 6-Sep-2026.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn A )   &    |- ( ph -> A. x e. A DECID ( F ` x ) R ( G ` x ) )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> DECID F oR R G )
27-Aug-2026	prmdcz 12933	Primality is decidable. (Contributed by Jim Kingdon, 27-Aug-2026.)
|- ( N e. ZZ -> DECID N e. Prime )
27-Aug-2026	zmincl 12028	The minumum of two integers is an integer. (Contributed by Jim Kingdon, 27-Aug-2026.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> inf ( { A , B } , RR , < ) e. ZZ )
25-Aug-2026	nn0sqdcq 13012	A nonnegative integer is a perfect square or not. This is similar to nn0sqdc 11167 but expresses the idea of being a perfect square as having a rational number which, when squared, gives the original number. (Contributed by Jim Kingdon, 25-Aug-2026.)
|- ( N e. NN0 -> DECID E. q e. QQ N = ( q ^ 2 ) )
25-Aug-2026	qabscl 11864	The absolute value of a rational number is a rational number. (Contributed by Jim Kingdon, 25-Aug-2026.)
|- ( A e. QQ -> ( abs ` A ) e. QQ )
25-Aug-2026	nn0sqdc 11167	A nonnegative integer is a perfect square or not. (Contributed by Jim Kingdon, 25-Aug-2026.)
|- ( N e. NN0 -> DECID E. q e. NN0 N = ( q ^ 2 ) )
24-Aug-2026	sqrtrirr 13013	The square root of a nonnegative integer is either rational or irrational. (Contributed by Jim Kingdon, 24-Aug-2026.)
|- ( A e. NN0 -> ( ( sqr ` A ) e. QQ \/ ( ( sqr ` A ) e. RR /\ A. q e. QQ ( sqr ` A ) # q ) ) )
21-Aug-2026	prmefexple 16325	Convert a bound on a power of a prime to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Jim Kingdon, 21-Aug-2026.)
|- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( A ^ N ) <_ B <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) )
20-Aug-2026	zprmlogbap 16235	The logarithm of a natural number to a prime base is either rational or irrational. The proof decomposes X into m e. NN and a e. NN0 such that X = ( ( B ^ a ) x. m ) (using nnmaxpw 12977). If m = 1 the logarithm is a, which is rational. If 1 < m then we can apply logbgcd1irrap 16223 to show that the logarithm is irrational. (Contributed by Jim Kingdon and Taylor Barrella, 20-Aug-2026.)
|- ( ( X e. NN /\ B e. Prime ) -> ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) )
20-Aug-2026	zprmlogbaplem3 16234	Lemma for zprmlogbap 16235. Decomposing a natural number into a power of a prime base and a factor not divisible by that prime. (Contributed by Jim Kingdon, 20-Aug-2026.)
|- J = { z e. NN | -. B || z }   &    |- F = ( x e. J , y e. NN0 |-> ( ( B ^ y ) x. x ) )   =>    |- ( ( X e. NN /\ B e. Prime ) -> E. m e. NN E. a e. NN0 ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) )
20-Aug-2026	zprmlogbaplem2 16233	Lemma for zprmlogbap 16235. The logarithm is either rational or irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)
|- ( ph -> B e. Prime )   &    |- ( ph -> M e. NN )   &    |- ( ph -> -. B || M )   &    |- ( ph -> A e. NN0 )   &    |- X = ( ( B ^ A ) x. M )   =>    |- ( ph -> ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) )
20-Aug-2026	zprmlogbaplem1 16232	Lemma for zprmlogbap 16235. Rearranging an expression involving logarithms. (Contributed by Jim Kingdon, 20-Aug-2026.)
|- ( ph -> B e. Prime )   &    |- ( ph -> M e. NN )   &    |- ( ph -> -. B || M )   &    |- ( ph -> A e. NN0 )   =>    |- ( ph -> ( B logb ( ( B ^ A ) x. M ) ) = ( A + ( B logb M ) ) )
20-Aug-2026	flaplelt 10729	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 20-Aug-2026.)
|- ( ( A e. QQ \/ ( A e. RR /\ A. q e. QQ A # q ) ) -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )
20-Aug-2026	flapcl 10727	The floor (greatest integer) function yields an integer when applied to a number which is either rational or irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)
|- ( ( A e. QQ \/ ( A e. RR /\ A. q e. QQ A # q ) ) -> ( |_ ` A ) e. ZZ )
20-Aug-2026	irraddap 10062	The sum of an irrational number and a rational number is irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)
|- ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) -> ( ( A + B ) e. RR /\ A. q e. QQ ( A + B ) # q ) )
19-Aug-2026	nnmaxpw 12977	The function F that decomposes a number into its "odd" and "even" parts, which is to say the largest power of a base and largest divisor of the number not divisible by that base, is a bijection from pairs of a nonnegative integer and a number not divisible by that base to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.) (Revised by Jim Kingdon, 19-Aug-2026.)
|- J = { z e. NN | -. B || z }   &    |- F = ( x e. J , y e. NN0 |-> ( ( B ^ y ) x. x ) )   =>    |- ( B e. ( ZZ>= ` 2 ) -> F : ( J X. NN0 ) -1-1-onto-> NN )
19-Aug-2026	nnmaxpwlemparts 12976	Lemma for nnmaxpw 12977. Decomposing a number into parts. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 19-Aug-2026.)
|- ( B e. ( ZZ>= ` 2 ) -> ( ( ( ( X e. NN /\ -. B || X ) /\ Y e. NN0 ) /\ A = ( ( B ^ Y ) x. X ) ) <-> ( A e. NN /\ ( X = ( A / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) ) /\ Y = ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) ) ) )
19-Aug-2026	nnmaxpwlemnfac 12975	Lemma for nnmaxpw 12977. Removing the powers of a base from a natural number produces a number not divisible by that base. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 19-Aug-2026.)
|- ( ( A e. NN /\ B e. ( ZZ>= ` 2 ) ) -> -. B || ( A / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) ) )
19-Aug-2026	nnmaxpwlemndvds 12974	Lemma for nnmaxpw 12977. A natural number is not divisible by one more than the highest power of a base which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) (Revised by Jim Kingdon, 19-Aug-2026.)
|- ( ( A e. NN /\ B e. ( ZZ>= ` 2 ) ) -> -. ( B ^ ( ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) + 1 ) ) || A )
19-Aug-2026	nnmaxpwlemdvds 12973	Lemma for nnmaxpw 12977. A natural number is divisible by the highest power of a base which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) (Revised by Jim Kingdon, 19-Aug-2026.)
|- ( ( A e. NN /\ B e. ( ZZ>= ` 2 ) ) -> ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) || A )
18-Aug-2026	nnmaxpwlemxy 12972	Lemma for nnmaxpw 12977. Another way of stating that decomposing a natural number into a power of a base and a number not divisible by that base is unique. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)
|- ( ph -> X e. NN )   &    |- ( ph -> B e. ( ZZ>= ` 2 ) )   &    |- ( ph -> -. B || X )   &    |- ( ph -> Y e. NN0 )   &    |- ( ph -> A = ( ( B ^ Y ) x. X ) )   =>    |- ( ph -> ( X = ( A / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) ) /\ Y = ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) )
18-Aug-2026	pwbdvdseu 12971	A natural number has a unique highest power of a base which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)
|- ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) -> E! m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
18-Aug-2026	pwbdvdseulemle 12970	Lemma for pwbdvdseu 12971. Powers of a base which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)
|- ( ph -> N e. NN )   &    |- ( ph -> A e. NN0 )   &    |- ( ph -> B e. NN0 )   &    |- ( ph -> P e. NN )   &    |- ( ph -> ( P ^ A ) || N )   &    |- ( ph -> -. ( P ^ ( B + 1 ) ) || N )   =>    |- ( ph -> A <_ B )
18-Aug-2026	pwbdvds 12969	A natural number has a highest power of a base which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)
|- ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
17-Aug-2026	pwbdvdslemn 12968	Lemma for pwbdvds 12969. If a natural number has some power of a base which does not divide it, there is a highest power of the base which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.) (Revised by Jim Kingdon, 17-Aug-2026.)
|- ( ph -> N e. NN )   &    |- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   &    |- ( ph -> -. ( B ^ A ) || N )   =>    |- ( ph -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
14-Aug-2026	reaplog 16114	Apartness and the real natural logarithm. (Contributed by Jim Kingdon, 14-Aug-2026.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A # B <-> ( log ` A ) # ( log ` B ) ) )
13-Aug-2026	efap1p 16022	If the exponential of a number is apart from one plus that number, the number is apart from zero. To some extent can be thought of as the converse of efgt1p 12487. (Contributed by Jim Kingdon, 13-Aug-2026.)
|- ( ( A e. RR /\ ( 1 + A ) # ( exp ` A ) ) -> A # 0 )
6-Aug-2026	relndmfv 5731	The value of a relation outside its domain is the empty set. (Contributed by Jim Kingdon, 6-Aug-2026.)
|- ( ( Rel F /\ -. A e. dom F ) -> ( F ` A ) = (/) )
1-Aug-2026	wexmiddifxy 17268	Being able to subtract an arbitrary finite set from a finite set and get a finite set is equivalent to weak excluded middle. By adding additional conditions we can get a theorem which does not need weak excluded middle, at diffifi 7204. (Contributed by Jim Kingdon, 1-Aug-2026.)
|- (WEXMID <-> A. x A. y ( ( x e. Fin /\ y e. Fin ) -> ( x \ y ) e. Fin ) )
1-Aug-2026	wexmiddifxylem 17267	Lemma for wexmiddifxylem 17267. Showing weak excluded middle given a suitable finite set. (Contributed by Jim Kingdon, 1-Aug-2026.)
|- ( ( { 1o } \ { { x e. 1o | ph } } ) e. Fin -> DECID -. ph )
31-Jul-2026	rabid1o 17256	Converting between propositions and corresponding subsets of a singleton. (Contributed by Jim Kingdon, 31-Jul-2026.)
|- ( { x e. 1o | ph } = 1o <-> ph )
30-Jul-2026	wexmiddc 17264	Weak excluded middle expressed using WEXMID implies decidability of a negated proposition. (Contributed by Jim Kingdon, 30-Jul-2026.)
|- (WEXMID -> DECID -. ph )
30-Jul-2026	df-wexmid 17263	Weak excluded middle is the principle that any negated proposition is decidable. (Contributed by Jim Kingdon, 30-Jul-2026.)
|- (WEXMID <-> A. p e. ~P 1o ( -. p = 1o \/ -. -. p = 1o ) )
29-Jul-2026	wexmiddiffi 17266	Being able to subtract an arbitrary set from a finite set and get a finite set is equivalent to weak excluded middle. By adding additional conditions we can get a theorem which does not need weak excluded middle, at diffifi 7204. (Contributed by Jim Kingdon, 29-Jul-2026.)
|- (WEXMID <-> A. x A. y ( x e. Fin -> ( x \ y ) e. Fin ) )
29-Jul-2026	wexmiddiffilem 17265	Lemma for wexmiddiffi 17266. The reverse direction, using different notation. (Contributed by Jim Kingdon, 29-Jul-2026.)
|- ( A. x A. y ( x e. Fin -> ( x \ y ) e. Fin ) -> ( -. ph \/ -. -. ph ) )
28-Jul-2026	psrbaglefifi 15175	There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.) (Revised by Jim Kingdon, 28-Jul-2026.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ I e. Fin ) -> { y e. D | y oR <_ F } e. Fin )
24-Jul-2026	stnot 17261	A proposition is double negation stable if and only if it is equivalent to a negated proposition. Here by "proposition" we mean a subset of a singleton (which is a choice which allows us to quantify over them). Posed as an exercise online by Yannick Forster. (Contributed by Jim Kingdon, 24-Jul-2026.)
|- ( A e. ~P 1o -> ( ( -. -. A = 1o -> A = 1o ) <-> E. y e. ~P 1o ( A = 1o <-> -. y = 1o ) ) )
24-Jul-2026	slotm 13472	A structure with an inhabited slot is inhabited. (Contributed by Jim Kingdon, 24-Jul-2026.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   =>    |- ( A e. ( E ` G ) -> E. j j e. G )
22-Jul-2026	alseu-no-surprise 17403	Demonstrate that there is never a "surprise" when using the "all some one" quantifier, that is, it is never possible for the consequent to be both always true and always false. This follows from als-no-surprise 17371 by alseuals 17389. See als-no-surprise 17371 for why ordinary "for all" with implication has no such property. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- -. ( A.E! x ( ph -> ps ) /\ A.E! x ( ph -> -. ps ) )
22-Jul-2026	alseueu 17402	"The ph is ps " implies that exactly one thing is both ph and ps. This is the half of dfalseu2 17401 that drops the universal conjunct; it does not reverse, so E! x ( ph /\ ps ) cannot be used in place of an "all some one" statement. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x ( ph -> ps ) -> E! x ( ph /\ ps ) )
22-Jul-2026	dfalseu2 17401	An "all some one" statement is equivalent to its universal part conjoined with the claim that exactly one x satisfies both ph and ps. In other words, given A. x ( ph -> ps ), requiring exactly one x to satisfy ph, which is what df-alseu 17386 requires, and requiring exactly one x to satisfy ( ph /\ ps ) come to the same thing. Read ph as "is a king" and ps as "is hungry": if every king is hungry, then "there is exactly one king" and "there is exactly one hungry king" say the same thing, so either of them, together with "every king is hungry", gives "the king is hungry". The universal conjunct is what makes that work, and it cannot be dropped. E! x ( ph /\ ps ) on its own is strictly weaker than A.E! x ( ph -> ps ), since it is satisfied when many things are ph and just one of those is ps, as in a region with five kings exactly one of whom is hungry; see alseueu 17402 for the one direction that does hold without it. Uniqueness attaches to the antecedent, not to the conjunction. Russell's analysis of a definite description is built the same way: its uniqueness clause constrains the description predicate alone, while the predication is a separate conjunct. See his worked example of "the father of Charles II was executed", [Russell1905] p. 482. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E! x ( ph /\ ps ) ) )
22-Jul-2026	nfralseu 17400	Bound-variable hypothesis builder for "all some one" restricted to a class. This is the "all some one" counterpart of nfrals 17369. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- F/_ x A   &    |- F/ x ph   &    |- F/ x ps   =>    |- F/ x A.E! y e. A ( ph -> ps )
22-Jul-2026	nfalseu 17399	Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 17368. Unlike the set.mm version of this theorem, no disjoint variable condition is needed, because nfeu 2105 here does not require one. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- F/ x ph   &    |- F/ x ps   =>    |- F/ x A.E! y ( ph -> ps )
22-Jul-2026	ralseubii 17398	Congruence for "all some one" restricted to a class. This is the "all some one" counterpart of ralsbii 17366. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph <-> ch )   &    |- ( ps <-> th )   =>    |- ( A.E! x e. A ( ph -> ps ) <-> A.E! x e. A ( ch -> th ) )
22-Jul-2026	alseubii 17397	Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 17365. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph <-> ch )   &    |- ( ps <-> th )   =>    |- ( A.E! x ( ph -> ps ) <-> A.E! x ( ch -> th ) )
22-Jul-2026	ralseu2d 17396	Deduction rule: Given "all some one" applied to a class, you can extract the "exactly one" part. Note that the witness must satisfy the antecedent ps, not merely be a member of A. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A.E! x e. A ( ps -> ch ) )   =>    |- ( ph -> E! x e. A ps )
22-Jul-2026	ralseu1d 17395	Deduction rule: Given "all some one" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A.E! x e. A ( ps -> ch ) )   =>    |- ( ph -> A. x e. A ( ps -> ch ) )
22-Jul-2026	alseu2d 17394	Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "exactly one" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A.E! x ( ps -> ch ) )   =>    |- ( ph -> E! x ps )
22-Jul-2026	alseu1d 17393	Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A.E! x ( ps -> ch ) )   =>    |- ( ph -> A. x ( ps -> ch ) )
22-Jul-2026	ralseud 17392	Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 17395 and ralseu2d 17396 taken together. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A. x e. A ( ps -> ch ) )   &    |- ( ph -> E! x e. A ps )   =>    |- ( ph -> A.E! x e. A ( ps -> ch ) )
22-Jul-2026	alseud 17391	Introduction rule: "all some one" holds if the "for all" part holds and the antecedent has exactly one witness. This is the converse of alseu1d 17393 and alseu2d 17394 taken together. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A. x ( ps -> ch ) )   &    |- ( ph -> E! x ps )   =>    |- ( ph -> A.E! x ( ps -> ch ) )
22-Jul-2026	ralseurals 17390	"All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals 17389. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x e. A ( ph -> ps ) -> A.E. x e. A ( ph -> ps ) )
22-Jul-2026	alseuals 17389	"All some one" implies "all some": requiring exactly one witness is stronger than requiring at least one. Any consequence of an allsome statement is therefore a consequence of the corresponding "all some one" statement, which is how alseu-no-surprise 17403 is proved. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x ( ph -> ps ) -> A.E. x ( ph -> ps ) )
22-Jul-2026	dfralseu2 17388	The bounded "all some one" form is the general form with the class membership folded into the antecedent. This is the "all some one" counterpart of dfrals2 17354. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x e. A ( ph -> ps ) <-> A.E! x ( ( x e. A /\ ph ) -> ps ) )
22-Jul-2026	df-ralseu 17387	Define "all some one" applied to a class, which means ps is true whenever ph is true for x in A, and exactly one x in A satisfies ph. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E! x e. A ph ) )
22-Jul-2026	df-alseu 17386	Define "all some one" applied to a top-level implication, which means ps is true whenever ph is true and exactly one x satisfies ph. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E! x ph ) )
22-Jul-2026	wralseu 17385	Extend wff definition to include "all some one" applied to a class, which means ps is true whenever ph is true for x in A, and exactly one x in A satisfies ph. (Contributed by David A. Wheeler, 22-Jul-2026.)
wff A.E! x e. A ( ph -> ps )
22-Jul-2026	walseu 17384	Extend wff definition to include "all some one" applied to a top-level implication, which means ps is true whenever ph is true, and exactly one x satisfies ph. (Contributed by David A. Wheeler, 22-Jul-2026.)
wff A.E! x ( ph -> ps )
22-Jul-2026	mptmex 5948	If a function given by maps-to notation is inhabited, then the class it is defined on is inhabited. (Contributed by Jim Kingdon, 22-Jul-2026.)
|- ( C e. ( x e. A |-> B ) -> E. y y e. A )
20-Jul-2026	2alsraln0idm 17383	Nested general "all some" quantifiers with class membership as their antecedents, for the same class A: ph holds for every x and every y in A, and A is inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( A.E. x ( x e. A -> A.E. y ( y e. A -> ph ) ) <-> ( A. x e. A A. y e. A ph /\ E. x x e. A ) )
20-Jul-2026	2alsraln0m 17382	Nested general "all some" quantifiers with class membership as their antecedents: ph holds for every x in A and every y in B, and both A and B are inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( A.E. x ( x e. A -> A.E. y ( y e. B -> ph ) ) <-> ( A. x e. A A. y e. B ph /\ ( E. x x e. A /\ E. y y e. B ) ) )
20-Jul-2026	n0alsm 17381	If A is inhabited, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that ph holds for every x in A. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( E. x x e. A -> ( A.E. x ( x e. A -> ph ) <-> A. x e. A ph ) )
20-Jul-2026	alsraln0m 17378	The general "all some" quantifier with class membership as its antecedent holds if and only if ph holds for every x in A and A is inhabited. This is the intuitionistic form of what set.mm states using A =/= (/); see the section comment. (Contributed by Peter Mazsa, 28-Nov-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( A.E. x ( x e. A -> ph ) <-> ( A. x e. A ph /\ E. x x e. A ) )
20-Jul-2026	alsralrex 17377	The general "all some" quantifier with class membership as its antecedent holds if and only if ph holds for every x in A and some x in A satisfies ph. (Contributed by Peter Mazsa, 27-Nov-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( A.E. x ( x e. A -> ph ) <-> ( A. x e. A ph /\ E. x e. A ph ) )
20-Jul-2026	ralsanmo 17376	An "all some" statement restricted to a class, conjoined with the claim that at most one x in A satisfies its antecedent, is equivalent to the universal part conjoined with the claim that exactly one x in A satisfies the antecedent. This is the restricted counterpart of alsanmo 17375. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
|- ( ( A.E. x e. A ( ph -> ps ) /\ E* x e. A ph ) <-> ( A. x e. A ( ph -> ps ) /\ E! x e. A ph ) )
20-Jul-2026	alsanmo 17375	An "all some" statement conjoined with the claim that at most one x satisfies its antecedent is equivalent to the universal part conjoined with the claim that exactly one x satisfies the antecedent. The "all some" quantifier supplies the existence of such an x and E* x ph supplies the at-most-one part, so together they yield E! x ph. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
|- ( ( A.E. x ( ph -> ps ) /\ E* x ph ) <-> ( A. x ( ph -> ps ) /\ E! x ph ) )
20-Jul-2026	rexrals 17374	If a member of A satisfying the antecedent exists, then a restricted "all some" statement reduces to its universal part. This is the restricted counterpart of rexals 17380. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
|- ( E. x e. A ph -> ( A.E. x e. A ( ph -> ps ) <-> A. x e. A ( ph -> ps ) ) )
20-Jul-2026	ralrals 17373	If the universal part of a restricted "all some" statement holds, then the statement reduces to the existence of a member of A satisfying its antecedent. This is the restricted counterpart of ralals 17379. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
|- ( A. x e. A ( ph -> ps ) -> ( A.E. x e. A ( ph -> ps ) <-> E. x e. A ph ) )