HomeHome Intuitionistic Logic Explorer
Theorem List (p. 68 of 132)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0dom 6701 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   =>    |-  (/)  ~<_  A
 
Theoremenen1 6702 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( A  ~~  C  <->  B 
 ~~  C ) )
 
Theoremenen2 6703 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( C  ~~  A  <->  C 
 ~~  B ) )
 
Theoremdomen1 6704 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( A  ~<_  C  <->  B  ~<_  C )
 )
 
Theoremdomen2 6705 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B )
 )
 
2.6.28  Equinumerosity (cont.)
 
Theoremxpf1o 6706* Construct a bijection on a Cartesian product given bijections on the factors. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  ( x  e.  A  |->  X ) : A -1-1-onto-> B )   &    |-  ( ph  ->  ( y  e.  C  |->  Y ) : C -1-1-onto-> D )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  C  |->  <. X ,  Y >. ) : ( A  X.  C ) -1-1-onto-> ( B  X.  D ) )
 
Theoremxpen 6707 Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
 
Theoremmapen 6708 Two set exponentiations are equinumerous when their bases and exponents are equinumerous. Theorem 6H(c) of [Enderton] p. 139. (Contributed by NM, 16-Dec-2003.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  ^m  C )  ~~  ( B 
 ^m  D ) )
 
Theoremmapdom1g 6709 Order-preserving property of set exponentiation. (Contributed by Jim Kingdon, 15-Jul-2022.)
 |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  ( A  ^m  C ) 
 ~<_  ( B  ^m  C ) )
 
Theoremmapxpen 6710 Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96. (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( ( A  ^m  B )  ^m  C ) 
 ~~  ( A  ^m  ( B  X.  C ) ) )
 
Theoremxpmapenlem 6711* Lemma for xpmapen 6712. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  =  ( z  e.  C  |->  ( 1st `  ( x `  z ) ) )   &    |-  R  =  ( z  e.  C  |->  ( 2nd `  ( x `  z ) ) )   &    |-  S  =  ( z  e.  C  |->  <.
 ( ( 1st `  y
 ) `  z ) ,  ( ( 2nd `  y
 ) `  z ) >. )   =>    |-  ( ( A  X.  B )  ^m  C ) 
 ~~  ( ( A 
 ^m  C )  X.  ( B  ^m  C ) )
 
Theoremxpmapen 6712 Equinumerosity law for set exponentiation of a Cartesian product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  X.  B )  ^m  C )  ~~  ( ( A  ^m  C )  X.  ( B  ^m  C ) )
 
Theoremssenen 6713* Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  ~~  B  ->  { x  |  ( x  C_  A  /\  x  ~~  C ) }  ~~  { x  |  ( x  C_  B  /\  x  ~~  C ) }
 )
 
2.6.29  Pigeonhole Principle
 
Theoremphplem1 6714 Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } )
 )
 
Theoremphplem2 6715 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremphplem3 6716 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6718. (Contributed by NM, 26-May-1998.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremphplem4 6717 Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  ~~ 
 suc  B  ->  A  ~~  B ) )
 
Theoremphplem3g 6718 A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6716 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremnneneq 6719 Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~~  B 
 <->  A  =  B ) )
 
Theoremphp5 6720 A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  om  ->  -.  A  ~~  suc  A )
 
Theoremsnnen2og 6721 A singleton  { A } is never equinumerous with the ordinal number 2. If  A is a proper class, see snnen2oprc 6722. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( A  e.  V  ->  -.  { A }  ~~  2o )
 
Theoremsnnen2oprc 6722 A singleton  { A } is never equinumerous with the ordinal number 2. If  A is a set, see snnen2og 6721. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( -.  A  e.  _V 
 ->  -.  { A }  ~~  2o )
 
Theorem1nen2 6723 One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
 |- 
 -.  1o  ~~  2o
 
Theoremphplem4dom 6724 Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  ~<_  suc  B  ->  A  ~<_  B ) )
 
Theoremphp5dom 6725 A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( A  e.  om  ->  -.  suc  A  ~<_  A )
 
Theoremnndomo 6726 Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~<_  B  <->  A  C_  B ) )
 
Theoremphpm 6727* Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols  E. x x  e.  ( A  \  B
) (which is stronger than not being equal in the absence of excluded middle). Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 6714 through phplem4 6717, nneneq 6719, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  C_  A  /\  E. x  x  e.  ( A  \  B ) )  ->  -.  A  ~~  B )
 
Theoremphpelm 6728 Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  A )  ->  -.  A  ~~  B )
 
Theoremphplem4on 6729 Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( suc  A  ~~ 
 suc  B  ->  A  ~~  B ) )
 
2.6.30  Finite sets
 
Theoremfict 6730 A finite set is dominated by  om. Also see finct 6969. (Contributed by Thierry Arnoux, 27-Mar-2018.)
 |-  ( A  e.  Fin  ->  A 
 ~<_  om )
 
Theoremfidceq 6731 Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that  { B ,  C } is finite would require showing it is equinumerous to  1o or to  2o but to show that you'd need to know  B  =  C or  -.  B  =  C, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  -> DECID  B  =  C )
 
Theoremfidifsnen 6732 All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
 |-  ( ( X  e.  Fin  /\  A  e.  X  /\  B  e.  X )  ->  ( X  \  { A } )  ~~  ( X  \  { B }
 ) )
 
Theoremfidifsnid 6733 If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3636 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  A ) 
 ->  ( ( A  \  { B } )  u. 
 { B } )  =  A )
 
Theoremnnfi 6734 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( A  e.  om  ->  A  e.  Fin )
 
Theoremenfi 6735 Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
 |-  ( A  ~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
 
Theoremenfii 6736 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( B  e.  Fin  /\  A  ~~  B ) 
 ->  A  e.  Fin )
 
Theoremssfilem 6737* Lemma for ssfiexmid 6738. (Contributed by Jim Kingdon, 3-Feb-2022.)
 |- 
 { z  e.  { (/)
 }  |  ph }  e.  Fin   =>    |-  ( ph  \/  -.  ph )
 
Theoremssfiexmid 6738* If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)
 |- 
 A. x A. y
 ( ( x  e. 
 Fin  /\  y  C_  x )  ->  y  e.  Fin )   =>    |-  ( ph  \/  -.  ph )
 
Theoreminfiexmid 6739* If the intersection of any finite set and any other set is finite, excluded middle follows. (Contributed by Jim Kingdon, 5-Feb-2022.)
 |-  ( x  e.  Fin  ->  ( x  i^i  y )  e.  Fin )   =>    |-  ( ph  \/  -.  ph )
 
Theoremdomfiexmid 6740* If any set dominated by a finite set is finite, excluded middle follows. (Contributed by Jim Kingdon, 3-Feb-2022.)
 |-  ( ( x  e. 
 Fin  /\  y  ~<_  x ) 
 ->  y  e.  Fin )   =>    |-  ( ph  \/  -.  ph )
 
Theoremdif1en 6741 If a set  A is equinumerous to the successor of a natural number  M, then  A with an element removed is equinumerous to  M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
 |-  ( ( M  e.  om 
 /\  A  ~~  suc  M 
 /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M )
 
Theoremdif1enen 6742 Subtracting one element from each of two equinumerous finite sets. (Contributed by Jim Kingdon, 5-Jun-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A 
 ~~  B )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  D  e.  B )   =>    |-  ( ph  ->  ( A  \  { C }
 )  ~~  ( B  \  { D } )
 )
 
Theoremfiunsnnn 6743 Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
 |-  ( ( ( A  e.  Fin  /\  B  e.  ( _V  \  A ) )  /\  ( N  e.  om  /\  A  ~~  N ) )  ->  ( A  u.  { B } )  ~~  suc  N )
 
Theoremphp5fin 6744 A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  ( _V  \  A ) )  ->  -.  A  ~~  ( A  u.  { B }
 ) )
 
Theoremfisbth 6745 Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
 |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~~  B )
 
Theorem0fin 6746 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
 |-  (/)  e.  Fin
 
Theoremfin0 6747* A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
 |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
 
Theoremfin0or 6748* A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
 |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  E. x  x  e.  A ) )
 
Theoremdiffitest 6749* If subtracting any set from a finite set gives a finite set, any proposition of the form  -.  ph is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove  A  e.  Fin  ->  ( A  \  B
)  e.  Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
 |- 
 A. a  e.  Fin  A. b ( a  \  b )  e.  Fin   =>    |-  ( -.  ph  \/  -.  -.  ph )
 
Theoremfindcard 6750* Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch ) )   &    |-  ( x  =  y  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  Fin  ->  (
 A. z  e.  y  ch  ->  th ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremfindcard2 6751* Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  Fin  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  Fin  ->  ta )
 
Theoremfindcard2s 6752* Variation of findcard2 6751 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 ( y  e.  Fin  /\ 
 -.  z  e.  y
 )  ->  ( ch  ->  th ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremfindcard2d 6753* Deduction version of findcard2 6751. If you also need  y  e.  Fin (which doesn't come for free due to ssfiexmid 6738), use findcard2sd 6754 instead. (Contributed by SO, 16-Jul-2018.)
 |-  ( x  =  (/)  ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  ( ps  <->  th ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ps  <->  ta ) )   &    |-  ( x  =  A  ->  ( ps  <->  et ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\  ( y  C_  A  /\  z  e.  ( A  \  y ) ) )  ->  ( th  ->  ta ) )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  et )
 
Theoremfindcard2sd 6754* Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
 |-  ( x  =  (/)  ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  ( ps  <->  th ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ps  <->  ta ) )   &    |-  ( x  =  A  ->  ( ps  <->  et ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( (
 ph  /\  y  e.  Fin )  /\  ( y 
 C_  A  /\  z  e.  ( A  \  y
 ) ) )  ->  ( th  ->  ta )
 )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  et )
 
Theoremdiffisn 6755 Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  A ) 
 ->  ( A  \  { B } )  e.  Fin )
 
Theoremdiffifi 6756 Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  B  C_  A )  ->  ( A  \  B )  e.  Fin )
 
Theoreminfnfi 6757 An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
 |-  ( om  ~<_  A  ->  -.  A  e.  Fin )
 
Theoremominf 6758 The set of natural numbers is not finite. Although we supply this theorem because we can, the more natural way to express " om is infinite" is  om  ~<_  om which is an instance of domrefg 6629. (Contributed by NM, 2-Jun-1998.)
 |- 
 -.  om  e.  Fin
 
Theoremisinfinf 6759* An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.)
 |-  ( om  ~<_  A  ->  A. n  e.  om  E. x ( x  C_  A  /\  x  ~~  n ) )
 
Theoremac6sfi 6760* Existence of a choice function for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( y  =  ( f `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  Fin  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremtridc 6761* A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)
 |-  ( ph  ->  R  Po  A )   &    |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  -> DECID  B R C )
 
Theoremfimax2gtrilemstep 6762* Lemma for fimax2gtri 6763. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.)
 |-  ( ph  ->  R  Po  A )   &    |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  U  C_  A )   &    |-  ( ph  ->  Z  e.  A )   &    |-  ( ph  ->  V  e.  A )   &    |-  ( ph  ->  -.  V  e.  U )   &    |-  ( ph  ->  A. y  e.  U  -.  Z R y )   =>    |-  ( ph  ->  E. x  e.  A  A. y  e.  ( U  u.  { V } )  -.  x R y )
 
Theoremfimax2gtri 6763* A finite set has a maximum under a trichotomous order. (Contributed by Jim Kingdon, 5-Sep-2022.)
 |-  ( ph  ->  R  Po  A )   &    |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/= 
 (/) )   =>    |-  ( ph  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremfinexdc 6764* Decidability of existence, over a finite set and defined by a decidable proposition. (Contributed by Jim Kingdon, 12-Jul-2022.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A DECID  ph )  -> DECID  E. x  e.  A  ph )
 
Theoremdfrex2fin 6765* Relationship between universal and existential quantifiers over a finite set. Remark in Section 2.2.1 of [Pierik], p. 8. Although Pierik does not mention the decidability condition explicitly, it does say "only finitely many x to check" which means there must be some way of checking each value of x. (Contributed by Jim Kingdon, 11-Jul-2022.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A DECID  ph )  ->  ( E. x  e.  A  ph  <->  -.  A. x  e.  A  -.  ph )
 )
 
Theoreminfm 6766* An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.)
 |-  ( om  ~<_  A  ->  E. x  x  e.  A )
 
Theoreminfn0 6767 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
 |-  ( om  ~<_  A  ->  A  =/=  (/) )
 
Theoreminffiexmid 6768* If any given set is either finite or infinite, excluded middle follows. (Contributed by Jim Kingdon, 15-Jun-2022.)
 |-  ( x  e.  Fin  \/ 
 om  ~<_  x )   =>    |-  ( ph  \/  -.  ph )
 
Theoremen2eqpr 6769 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C ) 
 ->  ( A  =/=  B  ->  C  =  { A ,  B } ) )
 
Theoremexmidpw 6770 Excluded middle is equivalent to the power set of  1o having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)
 |-  (EXMID  <->  ~P 1o  ~~  2o )
 
Theoremfientri3 6771 Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  ~<_  B  \/  B 
 ~<_  A ) )
 
Theoremnnwetri 6772* A natural number is well-ordered by 
_E. More specifically, this order both satisfies  We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
 |-  ( A  e.  om  ->  (  _E  We  A  /\  A. x  e.  A  A. y  e.  A  ( x  _E  y  \/  x  =  y  \/  y  _E  x ) ) )
 
Theoremonunsnss 6773 Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
 |-  ( ( B  e.  V  /\  ( A  u.  { B } )  e. 
 On )  ->  B  C_  A )
 
Theoremunfiexmid 6774* If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.)
 |-  ( ( x  e. 
 Fin  /\  y  e.  Fin )  ->  ( x  u.  y )  e.  Fin )   =>    |-  ( ph  \/  -.  ph )
 
Theoremunsnfi 6775 Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  V  /\  -.  B  e.  A ) 
 ->  ( A  u.  { B } )  e.  Fin )
 
Theoremunsnfidcex 6776 The  B  e.  V condition in unsnfi 6775. This is intended to show that unsnfi 6775 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  -.  B  e.  A  /\  ( A  u.  { B } )  e.  Fin )  -> DECID  -.  B  e.  _V )
 
Theoremunsnfidcel 6777 The  -.  B  e.  A condition in unsnfi 6775. This is intended to show that unsnfi 6775 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  V  /\  ( A  u.  { B } )  e.  Fin )  -> DECID  -.  B  e.  A )
 
Theoremunfidisj 6778 The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  e. 
 Fin )
 
Theoremundifdcss 6779* Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.)
 |-  ( A  =  ( B  u.  ( A 
 \  B ) )  <-> 
 ( B  C_  A  /\  A. x  e.  A DECID  x  e.  B ) )
 
Theoremundifdc 6780* Union of complementary parts into whole. This is a case where we can strengthen undifss 3413 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.)
 |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  B  e.  Fin  /\  B  C_  A )  ->  A  =  ( B  u.  ( A  \  B ) ) )
 
Theoremundiffi 6781 Union of complementary parts into whole. This is a case where we can strengthen undifss 3413 from subset to equality. (Contributed by Jim Kingdon, 2-Mar-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  B  C_  A )  ->  A  =  ( B  u.  ( A  \  B ) ) )
 
Theoremunfiin 6782 The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  e.  Fin )  ->  ( A  u.  B )  e.  Fin )
 
Theoremprfidisj 6783 A pair is finite if it consists of two unequal sets. For the case where  A  =  B, see snfig 6676. For the cases where one or both is a proper class, see prprc1 3601, prprc2 3602, or prprc 3603. (Contributed by Jim Kingdon, 31-May-2022.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B ) 
 ->  { A ,  B }  e.  Fin )
 
Theoremtpfidisj 6784 A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  A  =/=  C )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  { A ,  B ,  C }  e.  Fin )
 
Theoremfiintim 6785* If a class is closed under pairwise intersections, then it is closed under nonempty finite intersections. The converse would appear to require an additional condition, such as  x and  y not being equal, or  A having decidable equality.

This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.)

 |-  ( A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A  ->  A. x ( ( x  C_  A  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  A ) )
 
Theoremxpfi 6786 The Cartesian product of two finite sets is finite. Lemma 8.1.16 of [AczelRathjen], p. 74. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  X.  B )  e.  Fin )
 
Theorem3xpfi 6787 The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( V  e.  Fin  ->  ( ( V  X.  V )  X.  V )  e.  Fin )
 
Theoremfisseneq 6788 A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
 |-  ( ( B  e.  Fin  /\  A  C_  B  /\  A  ~~  B )  ->  A  =  B )
 
Theoremphpeqd 6789 Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6727 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B 
 C_  A )   &    |-  ( ph  ->  A  ~~  B )   =>    |-  ( ph  ->  A  =  B )
 
Theoremssfirab 6790* A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A. x  e.  A DECID  ps )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  e.  Fin )
 
Theoremssfidc 6791* A subset of a finite set is finite if membership in the subset is decidable. (Contributed by Jim Kingdon, 27-May-2022.)
 |-  ( ( A  e.  Fin  /\  B  C_  A  /\  A. x  e.  A DECID  x  e.  B )  ->  B  e.  Fin )
 
Theoremsnon0 6792 An ordinal which is a singleton is  { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
 |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  =  (/) )
 
Theoremfnfi 6793 A version of fnex 5610 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
 
Theoremfundmfi 6794 The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  Fun  A )  ->  dom  A  e.  Fin )
 
Theoremfundmfibi 6795 A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.)
 |-  ( Fun  F  ->  ( F  e.  Fin  <->  dom  F  e.  Fin ) )
 
Theoremresfnfinfinss 6796 The restriction of a function to a finite subset of its domain is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
 |-  ( ( F  Fn  A  /\  B  e.  Fin  /\  B  C_  A )  ->  ( F  |`  B )  e.  Fin )
 
Theoremrelcnvfi 6797 If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)
 |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' A  e.  Fin )
 
Theoremfunrnfi 6798 The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)
 |-  ( ( Rel  A  /\  Fun  `' A  /\  A  e.  Fin )  ->  ran  A  e.  Fin )
 
Theoremf1ofi 6799 If a 1-1 and onto function has a finite domain, its range is finite. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  F : A -1-1-onto-> B )  ->  B  e.  Fin )
 
Theoremf1dmvrnfibi 6800 A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6801. (Contributed by AV, 10-Jan-2020.)
 |-  ( ( A  e.  V  /\  F : A -1-1-> B )  ->  ( F  e.  Fin  <->  ran  F  e.  Fin ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13177
  Copyright terms: Public domain < Previous  Next >