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Theorem List for Metamath Proof Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreminfunabs 7801 An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A  /\  B 
 ~<_  A )  ->  ( A  u.  B )  ~~  A )
 
Theoreminfcda 7802 The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\  om  ~<_  A )  ->  ( A  +c  B ) 
 ~~  ( A  u.  B ) )
 
Theoreminfdif 7803 The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B ) 
 ~~  A )
 
Theoreminfdif2 7804 Cardinality ordering for an infinite set difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\  om  ~<_  A )  ->  ( ( A  \  B )  ~<_  B  <->  A  ~<_  B )
 )
 
Theoreminfxpdom 7805 Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A  /\  B 
 ~<_  A )  ->  ( A  X.  B )  ~<_  A )
 
Theoreminfxpabs 7806 Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  ->  ( A  X.  B )  ~~  A )
 
Theoreminfunsdom1 7807 The union of two sets that are strictly dominated by the infinite set  X is also dominated by  X. This version of infunsdom 7808 assumes additionally that  A is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
 
Theoreminfunsdom 7808 The union of two sets that are strictly dominated by the infinite set  X is also strictly dominated by  X. (Contributed by Mario Carneiro, 3-May-2015.)
 |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A 
 ~<  X  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
 
Theoreminfxp 7809 Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/= 
 (/) ) )  ->  ( A  X.  B ) 
 ~~  ( A  u.  B ) )
 
Theorempwcdadom 7810 A property of dominance over a powerset, and a main lemma for gchac 8263. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ~P ( A  +c  A )  ~<_  ( A  +c  B ) 
 ->  ~P A  ~<_  B )
 
Theoreminfpss 7811* Every infinite set has an equinumerous proper subset. Exercise 7 of [TakeutiZaring] p. 91. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( om  ~<_  A  ->  E. x ( x  C.  A  /\  x  ~~  A ) )
 
Theoreminfmap2 7812* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8166 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( om  ~<_  A  /\  B 
 ~<_  A  /\  ( A 
 ^m  B )  e. 
 dom  card )  ->  ( A  ^m  B )  ~~  { x  |  ( x 
 C_  A  /\  x  ~~  B ) } )
 
2.6.10  The Ackermann bijection
 
Theoremackbij2lem1 7813 Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  om  ->  ~P A  C_  ( ~P om  i^i  Fin )
 )
 
Theoremackbij1lem1 7814 Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( -.  A  e.  B  ->  ( B  i^i  suc 
 A )  =  ( B  i^i  A ) )
 
Theoremackbij1lem2 7815 Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  B  ->  ( B  i^i  suc  A )  =  ( { A }  u.  ( B  i^i  A ) ) )
 
Theoremackbij1lem3 7816 Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  om  ->  A  e.  ( ~P
 om  i^i  Fin ) )
 
Theoremackbij1lem4 7817 Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  ( A  e.  om  ->  { A }  e.  ( ~P om  i^i  Fin ) )
 
Theoremackbij1lem5 7818 Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  ( A  e.  om  ->  ( card `  ~P suc  A )  =  ( ( card `  ~P A )  +o  ( card `  ~P A ) ) )
 
Theoremackbij1lem6 7819 Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  e.  ( ~P om  i^i  Fin )
 )  ->  ( A  u.  B )  e.  ( ~P om  i^i  Fin )
 )
 
Theoremackbij1lem7 7820* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  ( ~P om  i^i  Fin )  ->  ( F `  A )  =  ( card ` 
 U_ y  e.  A  ( { y }  X.  ~P y ) ) )
 
Theoremackbij1lem8 7821* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  om  ->  ( F `  { A } )  =  ( card `  ~P A ) )
 
Theoremackbij1lem9 7822* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  e.  ( ~P om  i^i  Fin )  /\  ( A  i^i  B )  =  (/) )  ->  ( F `  ( A  u.  B ) )  =  ( ( F `
  A )  +o  ( F `  B ) ) )
 
Theoremackbij1lem10 7823* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  F : ( ~P om  i^i  Fin ) --> om
 
Theoremackbij1lem11 7824* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  C_  A )  ->  B  e.  ( ~P om  i^i  Fin )
 )
 
Theoremackbij1lem12 7825* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( B  e.  ( ~P om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  A )  C_  ( F `
  B ) )
 
Theoremackbij1lem13 7826* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( F `  (/) )  =  (/)
 
Theoremackbij1lem14 7827* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
 
Theoremackbij1lem15 7828* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  e.  ( ~P om  i^i  Fin ) )  /\  (
 c  e.  om  /\  c  e.  A  /\  -.  c  e.  B ) )  ->  -.  ( F `  ( A  i^i  suc  c ) )  =  ( F `  ( B  i^i  suc  c )
 ) )
 
Theoremackbij1lem16 7829* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  e.  ( ~P om  i^i  Fin )
 )  ->  ( ( F `  A )  =  ( F `  B )  ->  A  =  B ) )
 
Theoremackbij1lem17 7830* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  F : ( ~P om  i^i  Fin ) -1-1-> om
 
Theoremackbij1lem18 7831* Lemma for ackbij1 7832. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  ( ~P om  i^i  Fin )  ->  E. b  e.  ( ~P om  i^i  Fin )
 ( F `  b
 )  =  suc  ( F `  A ) )
 
Theoremackbij1 7832* The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  F : ( ~P om  i^i  Fin ) -1-1-onto-> om
 
Theoremackbij1b 7833* The Ackermann bijection, part 1b: the bijection from ackbij1 7832 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  om  ->  ( F " ~P A )  =  ( card ` 
 ~P A ) )
 
Theoremackbij2lem2 7834* Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   &    |-  G  =  ( x  e.  _V  |->  ( y  e. 
 ~P dom  x  |->  ( F `  ( x
 " y ) ) ) )   =>    |-  ( A  e.  om  ->  ( rec ( G ,  (/) ) `  A ) : ( R1 `  A ) -1-1-onto-> ( card `  ( R1 `  A ) ) )
 
Theoremackbij2lem3 7835* Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   &    |-  G  =  ( x  e.  _V  |->  ( y  e. 
 ~P dom  x  |->  ( F `  ( x
 " y ) ) ) )   =>    |-  ( A  e.  om  ->  ( rec ( G ,  (/) ) `  A )  C_  ( rec ( G ,  (/) ) `  suc  A ) )
 
Theoremackbij2lem4 7836* Lemma for ackbij2 7837. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   &    |-  G  =  ( x  e.  _V  |->  ( y  e. 
 ~P dom  x  |->  ( F `  ( x
 " y ) ) ) )   =>    |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  A ) )
 
Theoremackbij2 7837* The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   &    |-  G  =  ( x  e.  _V  |->  ( y  e. 
 ~P dom  x  |->  ( F `  ( x
 " y ) ) ) )   &    |-  H  =  U. ( rec ( G ,  (/) ) " om )   =>    |-  H : U. ( R1 " om ) -1-1-onto-> om
 
Theoremr1om 7838 The set of hereditarily finite sets is countable. See ackbij2 7837 for an explicit bijection that works without Infinity. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( R1 `  om )  ~~  om
 
Theoremfictb 7839 A set is countable iff its collection of finite intersections is countable. (Contributed by Jeff Hankins, 24-Aug-2009.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  B  ->  ( A  ~<_  om  <->  ( fi `  A )  ~<_  om )
 )
 
2.6.11  Cofinality (without Axiom of Choice)
 
Theoremcflem 7840* A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set  A. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  e.  V  ->  E. x E. y
 ( x  =  (
 card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) )
 
Theoremcfval 7841* Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number  A is the cardinality (size) of the smallest unbounded subset  y of the ordinal number. Unbounded means that for every member of  A, there is a member of  y that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( A  e.  On  ->  ( cf `  A )  =  |^| { x  |  E. y ( x  =  ( card `  y
 )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
 
Theoremcff 7842 Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |- 
 cf : On --> On
 
Theoremcfub 7843* An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( cf `  A )  C_  |^| { x  |  E. y ( x  =  ( card `  y )  /\  ( y  C_  A  /\  A  C_  U. y ) ) }
 
Theoremcflm 7844* Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)
 |-  ( ( A  e.  B  /\  Lim  A )  ->  ( cf `  A )  =  |^| { x  |  E. y ( x  =  ( card `  y
 )  /\  ( y  C_  A  /\  A  =  U. y ) ) }
 )
 
Theoremcf0 7845 Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)
 |-  ( cf `  (/) )  =  (/)
 
Theoremcardcf 7846 Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( card `  ( cf `  A ) )  =  ( cf `  A )
 
Theoremcflecard 7847 Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( cf `  A )  C_  ( card `  A )
 
Theoremcfle 7848 Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( cf `  A )  C_  A
 
Theoremcfon 7849 The cofinality of any set is an ordinal (although it only makes sense when  A is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  ( cf `  A )  e.  On
 
Theoremcfeq0 7850 Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
 |-  ( A  e.  On  ->  ( ( cf `  A )  =  (/)  <->  A  =  (/) ) )
 
Theoremcfsuc 7851 Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
 |-  ( A  e.  On  ->  ( cf `  suc  A )  =  1o )
 
Theoremcff1 7852* There is always a map from  ( cf `  A
) to  A (this is a stronger condition than the definition, which only presupposes a map from some  y  ~~  ( cf `  A ). (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( A  e.  On  ->  E. f ( f : ( cf `  A ) -1-1-> A  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z  C_  ( f `
  w ) ) )
 
Theoremcfflb 7853* If there is a cofinal map from  B to  A, then  B is at least  ( cf `  A
). This theorem and cff1 7852 motivate the picture of  ( cf `  A
) as the greatest lower bound of the domain of cofinal maps into  A. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f
 ( f : B --> A  /\  A. z  e.  A  E. w  e.  B  z  C_  (
 f `  w )
 )  ->  ( cf `  A )  C_  B ) )
 
Theoremcfval2 7854* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( A  e.  On  ->  ( cf `  A )  =  |^|_ x  e. 
 { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  ( card `  x )
 )
 
Theoremcoflim 7855* A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  =  A 
 <-> 
 A. x  e.  A  E. y  e.  B  x  C_  y ) )
 
Theoremcflim3 7856* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  A  e.  _V   =>    |-  ( Lim  A  ->  ( cf `  A )  =  |^|_ x  e. 
 { x  e.  ~P A  |  U. x  =  A }  ( card `  x ) )
 
Theoremcflim2 7857 The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  A  e.  _V   =>    |-  ( Lim  A  <->  Lim  ( cf `  A ) )
 
Theoremcfom 7858 Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Proof shortened by Mario Carneiro, 11-Jun-2015.)
 |-  ( cf `  om )  =  om
 
Theoremcfss 7859* There is a cofinal subset of  A of cardinality  ( cf `  A
). (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( Lim  A  ->  E. x ( x 
 C_  A  /\  x  ~~  ( cf `  A )  /\  U. x  =  A ) )
 
Theoremcfslb 7860 Any cofinal subset of  A is at least as large as  ( cf `  A
). (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( Lim 
 A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A ) 
 ~<_  B )
 
Theoremcfslbn 7861 Any subset of  A smaller than its cofinality has union less than  A. (This is the contrapositive to cfslb 7860.) (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( Lim 
 A  /\  B  C_  A  /\  B  ~<  ( cf `  A ) )  ->  U. B  e.  A )
 
Theoremcfslb2n 7862* Any small collection of small subsets of  A cannot have union  A, where "small" means smaller than the cofinality. This is a stronger version of cfslb 7860. This is a common application of cofinality: under AC,  ( aleph `  1
) is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( Lim 
 A  /\  A. x  e.  B  ( x  C_  A  /\  x  ~<  ( cf `  A ) ) ) 
 ->  ( B  ~<  ( cf `  A )  ->  U. B  =/=  A ) )
 
Theoremcofsmo 7863* Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by by Mario Carneiro, 20-Mar-2013.)
 |-  C  =  { y  e.  B  |  A. w  e.  y  ( f `  w )  e.  (
 f `  y ) }   &    |-  K  =  |^| { x  e.  B  |  z  C_  ( f `  x ) }   &    |-  O  = OrdIso (  _E  ,  C )   =>    |-  ( ( Ord 
 A  /\  B  e.  On )  ->  ( E. f ( f : B --> A  /\  A. z  e.  A  E. w  e.  B  z  C_  (
 f `  w )
 )  ->  E. x  e.  suc  B E. g
 ( g : x --> A  /\  Smo  g  /\  A. z  e.  A  E. v  e.  x  z  C_  ( g `  v
 ) ) ) )
 
Theoremcfsmolem 7864* Lemma for cfsmo 7865. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  F  =  ( z  e.  _V  |->  ( ( g `  dom  z
 )  u.  U_ t  e.  dom  z  suc  (
 z `  t )
 ) )   &    |-  G  =  (recs ( F )  |`  ( cf `  A ) )   =>    |-  ( A  e.  On  ->  E. f ( f : ( cf `  A )
 --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z 
 C_  ( f `  w ) ) )
 
Theoremcfsmo 7865* The map in cff1 7852 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( A  e.  On  ->  E. f ( f : ( cf `  A )
 --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z 
 C_  ( f `  w ) ) )
 
Theoremcfcoflem 7866* Lemma for cfcof 7868, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f
 ( f : B --> A  /\  Smo  f  /\  A. x  e.  A  E. y  e.  B  x  C_  ( f `  y
 ) )  ->  ( cf `  A )  C_  ( cf `  B ) ) )
 
Theoremcoftr 7867* If there is a cofinal map from  B to  A and another from  C to  A, then there is also a cofinal map from  C to  B. Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 7868. (Contributed by Mario Carneiro, 16-Mar-2013.)
 |-  H  =  ( t  e.  C  |->  |^| { n  e.  B  |  ( g `
  t )  C_  ( f `  n ) } )   =>    |-  ( E. f ( f : B --> A  /\  Smo  f  /\  A. x  e.  A  E. y  e.  B  x  C_  (
 f `  y )
 )  ->  ( E. g ( g : C --> A  /\  A. z  e.  A  E. w  e.  C  z  C_  (
 g `  w )
 )  ->  E. h ( h : C --> B  /\  A. s  e.  B  E. w  e.  C  s  C_  ( h `  w ) ) ) )
 
Theoremcfcof 7868* If there is a cofinal map from  A to  B, then they have the same cofinality. This was used as Definition 11.1 of [TakeutiZaring] p. 100, who defines an equivalence relation cof  ( A ,  B ) and defines our  cf ( B ) as the minimum  B such that cof  ( A ,  B
). (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f
 ( f : B --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  B  z  C_  ( f `  w ) )  ->  ( cf `  A )  =  (
 cf `  B )
 ) )
 
Theoremcfidm 7869 The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( cf `  ( cf `  A ) )  =  ( cf `  A )
 
Theoremalephsing 7870 The cofinality of a limit aleph is the same as the cofinality of its argument, so if  ( aleph `  A )  <  A, then  ( aleph `  A
) is singular. Conversely, if  ( aleph `  A ) is regular (i.e. weakly inaccessible), then  ( aleph `  A )  =  A, so  A has to be rather large (see alephfp 7703). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  ( Lim  A  ->  (
 cf `  ( aleph `  A ) )  =  ( cf `  A ) )
 
2.6.12  Eight inequivalent definitions of finite set
 
Theoremsornom 7871* The range of a single-step monotone function from  om into a partially ordered set is a chain. (Contributed by Stefan O'Rear, 3-Nov-2014.)
 |-  ( ( F  Fn  om 
 /\  A. a  e.  om  ( ( F `  a ) R ( F `  suc  a
 )  \/  ( F `
  a )  =  ( F `  suc  a ) )  /\  R  Po  ran  F )  ->  R  Or  ran  F )
 
Syntaxcfin1a 7872 Extend class notation to include the class of Ia-finite sets.
 class FinIa
 
Syntaxcfin2 7873 Extend class notation to include the class of II-finite sets.
 class FinII
 
Syntaxcfin4 7874 Extend class notation to include the class of IV-finite sets.
 class FinIV
 
Syntaxcfin3 7875 Extend class notation to include the class of III-finite sets.
 class FinIII
 
Syntaxcfin5 7876 Extend class notation to include the class of V-finite sets.
 class FinV
 
Syntaxcfin6 7877 Extend class notation to include the class of VI-finite sets.
 class FinVI
 
Syntaxcfin7 7878 Extend class notation to include the class of VII-finite sets.
 class FinVII
 
Definitiondf-fin1a 7879* A set is Ia-finite iff it is not the union of two I-infinite sets. Equivalent to definition Ia of [Levy58] p. 2. A I-infinite Ia-finite set is also known as an amorphous set. This is the second of Levy's eight definitions of finite set. Levy's I-finite is equivalent to our df-fin 6835 and not repeated here. These eight definitions are equivalent with Choice but strictly decreasing in strength in models where Choice fails; conversely, they provide a series of increasingly stronger notions of infiniteness. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinIa  =  { x  |  A. y  e.  ~P  x ( y  e.  Fin  \/  ( x  \  y
 )  e.  Fin ) }
 
Definitiondf-fin2 7880* A set is II-finite (Tarski finite) iff every nonempty chain of subsets contains a maximum element. Definition II of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinII  =  { x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
 )  ->  U. y  e.  y ) }
 
Definitiondf-fin4 7881* A set is IV-finite (Dedekind finite) iff it has no equinumerous proper subset. Definition IV of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinIV  =  { x  |  -.  E. y ( y  C.  x  /\  y  ~~  x ) }
 
Definitiondf-fin3 7882 A set is III-finite (weakly Dedekind finite) iff its power set is Dedekind finite. Definition III of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinIII  =  { x  |  ~P x  e. FinIV }
 
Definitiondf-fin5 7883 A set is V-finite iff it behaves finitely under  +c. Definition V of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinV  =  { x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x ) ) }
 
Definitiondf-fin6 7884 A set is VI-finite iff it behaves finitely under  X.. Definition VI of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinVI  =  { x  |  ( x  ~<  2o  \/  x  ~<  ( x  X.  x ) ) }
 
Definitiondf-fin7 7885* A set is VII-finite iff it cannot be infinitely well ordered. Equivalent to definition VII of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinVII  =  { x  |  -.  E. y  e.  ( On  \  om ) x  ~~  y }
 
Theoremisfin1a 7886* Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinIa  <->  A. y  e.  ~P  A ( y  e. 
 Fin  \/  ( A  \  y )  e.  Fin ) ) )
 
Theoremfin1ai 7887 Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( ( A  e. FinIa  /\  X  C_  A )  ->  ( X  e.  Fin  \/  ( A  \  X )  e.  Fin ) )
 
Theoremisfin2 7888* Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinII  <->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
 )  ->  U. y  e.  y ) ) )
 
Theoremfin2i 7889 Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( ( ( A  e. FinII  /\  B  C_  ~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  U. B  e.  B )
 
Theoremisfin3 7890 Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e. FinIII  <->  ~P A  e. FinIV )
 
Theoremisfin4 7891* Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinIV  <->  -.  E. y ( y  C.  A  /\  y  ~~  A ) ) )
 
Theoremfin4i 7892 Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( ( X  C.  A  /\  X  ~~  A )  ->  -.  A  e. FinIV )
 
Theoremisfin5 7893 Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e. FinV  <->  ( A  =  (/) 
 \/  A  ~<  ( A  +c  A ) ) )
 
Theoremisfin6 7894 Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e. FinVI  <->  ( A  ~<  2o 
 \/  A  ~<  ( A  X.  A ) ) )
 
Theoremisfin7 7895* Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinVII  <->  -.  E. y  e.  ( On  \  om ) A  ~~  y ) )
 
Theoremsdom2en01 7896 A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( A  ~<  2o  <->  ( A  =  (/) 
 \/  A  ~~  1o ) )
 
Theoreminfpssrlem1 7897 Lemma for infpssr 7902. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  F : B -1-1-onto-> A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   &    |-  G  =  ( rec ( `' F ,  C )  |`  om )   =>    |-  ( ph  ->  ( G `  (/) )  =  C )
 
Theoreminfpssrlem2 7898 Lemma for infpssr 7902. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  F : B -1-1-onto-> A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   &    |-  G  =  ( rec ( `' F ,  C )  |`  om )   =>    |-  ( M  e.  om  ->  ( G `  suc  M )  =  ( `' F `  ( G `
  M ) ) )
 
Theoreminfpssrlem3 7899 Lemma for infpssr 7902. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  F : B -1-1-onto-> A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   &    |-  G  =  ( rec ( `' F ,  C )  |`  om )   =>    |-  ( ph  ->  G : om --> A )
 
Theoreminfpssrlem4 7900 Lemma for infpssr 7902. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  F : B -1-1-onto-> A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   &    |-  G  =  ( rec ( `' F ,  C )  |`  om )   =>    |-  ( ( ph  /\  M  e.  om  /\  N  e.  M )  ->  ( G `
  M )  =/=  ( G `  N ) )
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