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Theorem List for Metamath Proof Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
2.6.7  Cardinal numbers
 
Syntaxccrd 7501 Extend class definition to include the cardinal size function.
 class  card
 
Syntaxcale 7502 Extend class definition to include the aleph function.
 class  aleph
 
Syntaxccf 7503 Extend class definition to include the cofinality function.
 class  cf
 
Syntaxwacn 7504 The axiom of choice for limited-length sequences.
 class AC  A
 
Definitiondf-card 7505* Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 8101 for its value, cardval2 7557 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 8106. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
 |- 
 card  =  ( x  e.  _V  |->  |^| { y  e. 
 On  |  y  ~~  x } )
 
Definitiondf-aleph 7506 Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as theorems aleph0 7626, alephsuc 7628, and alephlim 7627. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it. (Contributed by NM, 21-Oct-2003.)
 |-  aleph  =  rec (har ,  om )
 
Definitiondf-cf 7507* Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 7806 for its value and a description. (Contributed by NM, 1-Apr-2004.)
 |- 
 cf  =  ( x  e.  On  |->  |^| { y  |  E. z ( y  =  ( card `  z
 )  /\  ( z  C_  x  /\  A. v  e.  x  E. u  e.  z  v  C_  u ) ) } )
 
Definitiondf-acn 7508* Define a local and length-limited version of the axiom of choice. The definition of the predicate 
X  e. AC  A is that for all families of nonempty subsets of  X indexed on  A (i.e. functions  A --> ~P X  \  { (/) }), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |- AC  A  =  { x  |  ( A  e.  _V  /\ 
 A. f  e.  (
 ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
 )  e.  ( f `
  y ) ) }
 
Theoremcardf2 7509* The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 20-Sep-2014.)
 |- 
 card : { x  |  E. y  e.  On  y  ~~  x } --> On
 
Theoremcardon 7510 The cardinal number of a set is an ordinal number. Proposition 10.6(1) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  ( card `  A )  e.  On
 
Theoremisnum2 7511* A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A )
 
Theoremisnumi 7512 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card
 )
 
Theoremennum 7513 Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  ~~  B  ->  ( A  e.  dom  card  <->  B  e.  dom  card ) )
 
Theoremfinnum 7514 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  Fin  ->  A  e.  dom  card )
 
Theoremonenon 7515 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  On  ->  A  e.  dom  card )
 
Theoremtskwe 7516* A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  { x  e. 
 ~P A  |  x  ~<  A }  C_  A )  ->  A  e.  dom  card
 )
 
Theoremxpnum 7517 The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( A  X.  B )  e.  dom  card
 )
 
Theoremcardval3 7518* An alternative definition of the value of  ( card `  A ) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( A  e.  dom  card 
 ->  ( card `  A )  =  |^| { x  e. 
 On  |  x  ~~  A } )
 
Theoremcardid2 7519 Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( A  e.  dom  card 
 ->  ( card `  A )  ~~  A )
 
Theoremisnum3 7520 A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  dom  card  <->  (
 card `  A )  ~~  A )
 
Theoremoncardval 7521* The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 8101, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  ( A  e.  On  ->  ( card `  A )  =  |^| { x  e. 
 On  |  x  ~~  A } )
 
Theoremoncardid 7522 Any ordinal number is equinumerous to its cardinal number. Unlike cardid 8102, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  On  ->  ( card `  A )  ~~  A )
 
Theoremcardonle 7523 The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
 |-  ( A  e.  On  ->  ( card `  A )  C_  A )
 
Theoremcard0 7524 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
 |-  ( card `  (/) )  =  (/)
 
Theoremcardidm 7525 The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( card `  ( card `  A ) )  =  ( card `  A )
 
Theoremoncard 7526* A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( E. x  A  =  ( card `  x )  <->  A  =  ( card `  A ) )
 
Theoremficardom 7527 The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.)
 |-  ( A  e.  Fin  ->  ( card `  A )  e.  om )
 
Theoremficardid 7528 A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)
 |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
 
Theoremcardnn 7529 The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( A  e.  om  ->  ( card `  A )  =  A )
 
Theoremcardnueq0 7530 The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( A  e.  dom  card 
 ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
 
Theoremcardne 7531 No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
 |-  ( A  e.  ( card `  B )  ->  -.  A  ~~  B )
 
Theoremcarden2a 7532 If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7533 are meant to replace carden 8106 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)
 |-  ( ( ( card `  A )  =  (
 card `  B )  /\  ( card `  A )  =/= 
 (/) )  ->  A  ~~  B )
 
Theoremcarden2b 7533 If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7532 are meant to replace carden 8106 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( A  ~~  B  ->  ( card `  A )  =  ( card `  B )
 )
 
Theoremcard1 7534* A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)
 |-  ( ( card `  A )  =  1o  <->  E. x  A  =  { x } )
 
Theoremcardsn 7535 A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)
 |-  ( A  e.  V  ->  ( card `  { A }
 )  =  1o )
 
Theoremcarddomi2 7536 Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8109, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  V )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A 
 ~<_  B ) )
 
Theoremsdomsdomcardi 7537 A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  ( A  ~<  ( card `  B )  ->  A  ~<  B )
 
Theoremcardlim 7538 An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  ( om  C_  ( card `  A )  <->  Lim  ( card `  A ) )
 
Theoremcardsdomelir 7539 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7540 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( A  e.  ( card `  B )  ->  A  ~<  B )
 
Theoremcardsdomel 7540 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)
 |-  ( ( A  e.  On  /\  B  e.  dom  card
 )  ->  ( A  ~<  B  <->  A  e.  ( card `  B ) ) )
 
Theoremiscard 7541* Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( ( card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
 
Theoremiscard2 7542* Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( ( card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e. 
 On  ( A  ~~  x  ->  A  C_  x ) ) )
 
Theoremcarddom2 7543 Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8109, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
 
Theoremharcard 7544 The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( card `  (har `  A ) )  =  (har `  A )
 
Theoremcardprclem 7545* Lemma for cardprc 7546. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  A  =  { x  |  ( card `  x )  =  x }   =>    |- 
 -.  A  e.  _V
 
Theoremcardprc 7546 The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310. In this proof (which does not use AC), we cannot use Cantor's construction canth3 8116 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7192 to construct (effectively)  ( aleph `  suc  A ) from  ( aleph `  A
), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)
 |- 
 { x  |  (
 card `  x )  =  x }  e/  _V
 
Theoremcarduni 7547* The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.)
 |-  ( A  e.  V  ->  ( A. x  e.  A  ( card `  x )  =  x  ->  (
 card `  U. A )  =  U. A ) )
 
Theoremcardiun 7548* The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)
 |-  ( A  e.  V  ->  ( A. x  e.  A  ( card `  B )  =  B  ->  (
 card `  U_ x  e.  A  B )  = 
 U_ x  e.  A  B ) )
 
Theoremcardennn 7549 If  A is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.)
 |-  ( ( A  ~~  B  /\  B  e.  om )  ->  ( card `  A )  =  B )
 
Theoremcardsucinf 7550 The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.)
 |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( card `  suc  A )  =  ( card `  A ) )
 
Theoremcardsucnn 7551 The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 7550. (Contributed by NM, 7-Nov-2008.)
 |-  ( A  e.  om  ->  ( card `  suc  A )  =  suc  ( card `  A ) )
 
Theoremcardom 7552 The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.)
 |-  ( card `  om )  = 
 om
 
Theoremcarden2 7553 Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 8106, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( ( card `  A )  =  ( card `  B )  <->  A 
 ~~  B ) )
 
Theoremcardsdom2 7554 A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( ( card `  A )  e.  ( card `  B )  <->  A 
 ~<  B ) )
 
Theoremdomtri2 7555 Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( A  ~<_  B 
 <->  -.  B  ~<  A ) )
 
Theoremnnsdomel 7556 Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  e.  B 
 <->  A  ~<  B )
 )
 
Theoremcardval2 7557* An alternate version of the value of the cardinal number of a set. Compare cardval 8101. This theorem could be used to give us a simpler definition of  card in place of df-card 7505. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
 |-  ( A  e.  dom  card 
 ->  ( card `  A )  =  { x  e.  On  |  x  ~<  A }
 )
 
Theoremisinffi 7558* An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7009 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. f  f : B -1-1-> A )
 
Theoremfidomtri 7559 Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  V ) 
 ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
 
Theoremfidomtri2 7560 Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 7-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  Fin )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
 
Theoremharsdom 7561 The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  dom  card 
 ->  A  ~<  (har `  A ) )
 
Theoremonsdom 7562* Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.) (Proof shortened by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  dom  card 
 ->  E. x  e.  On  A  ~<  x )
 
Theoremharval2 7563* An alternative expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  dom  card 
 ->  (har `  A )  =  |^| { x  e. 
 On  |  A  ~<  x } )
 
Theoremcardmin2 7564* The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)
 |-  ( E. x  e. 
 On  A  ~<  x  <->  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
 
Theorempm54.43lem 7565* In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 7534), so that their  A  e.  1 means, in our notation,  A  e.  {
x  |  ( card `  x )  =  1o }. Here we show that this is equivalent to  A  ~~  1o so that we can use the latter more convenient notation in pm54.43 7566. (Contributed by NM, 4-Nov-2013.)
 |-  ( A  ~~  1o  <->  A  e.  { x  |  (
 card `  x )  =  1o } )
 
Theorempm54.43 7566 Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2.

Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 7534), so that their  A  e.  1 means, in our notation,  A  e.  { x  |  (
card `  x )  =  1o } which is the same as  A  ~~  1o by pm54.43lem 7565. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.)

Theorem pm110.643 7736 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)

 |-  ( ( A  ~~  1o  /\  B  ~~  1o )  ->  ( ( A  i^i  B )  =  (/) 
 <->  ( A  u.  B )  ~~  2o ) )
 
Theorempr2nelem 7567 Lemma for pr2ne 7568. (Contributed by FL, 17-Aug-2008.)
 |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B ) 
 ->  { A ,  B }  ~~  2o )
 
Theorempr2ne 7568 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  <->  A  =/=  B ) )
 
Theoremprdom2 7569 An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { A ,  B }  ~<_  2o )
 
Theoremen2eqpr 7570 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 } ) )
 
Theoremdif1card 7571 The cardinality of a non-empty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.)
 |-  ( ( A  e.  Fin  /\  X  e.  A ) 
 ->  ( card `  A )  =  suc  ( card `  ( A  \  { X }
 ) ) )
 
Theoremleweon 7572* Lexicographical order is a well-ordering of  On 
X.  On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7573, this order is not set-like, as the preimage of  <. 1o ,  (/) >. is the proper class  ( { (/) }  X.  On ). (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  L  =  { <. x ,  y >.  |  ( ( x  e.  ( On  X.  On )  /\  y  e.  ( On  X. 
 On ) )  /\  ( ( 1st `  x )  e.  ( 1st `  y )  \/  (
 ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x )  e.  ( 2nd `  y ) ) ) ) }   =>    |-  L  We  ( On  X.  On )
 
Theoremr0weon 7573* A set-like well ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  L  =  { <. x ,  y >.  |  ( ( x  e.  ( On  X.  On )  /\  y  e.  ( On  X. 
 On ) )  /\  ( ( 1st `  x )  e.  ( 1st `  y )  \/  (
 ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x )  e.  ( 2nd `  y ) ) ) ) }   &    |-  R  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z
 )  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w )  u.  ( 2nd `  w ) )  \/  (
 ( ( 1st `  z
 )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }   =>    |-  ( R  We  ( On  X.  On )  /\  R Se  ( On  X.  On ) )
 
Theoreminfxpenlem 7574* Lemma for infxpen 7575. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  L  =  { <. x ,  y >.  |  ( ( x  e.  ( On  X.  On )  /\  y  e.  ( On  X. 
 On ) )  /\  ( ( 1st `  x )  e.  ( 1st `  y )  \/  (
 ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x )  e.  ( 2nd `  y ) ) ) ) }   &    |-  R  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z
 )  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w )  u.  ( 2nd `  w ) )  \/  (
 ( ( 1st `  z
 )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }   &    |-  Q  =  ( R  i^i  ( ( a  X.  a )  X.  ( a  X.  a ) ) )   &    |-  ( ph  <->  ( ( a  e.  On  /\  A. m  e.  a  ( om  C_  m  ->  ( m  X.  m )  ~~  m ) )  /\  ( om  C_  a  /\  A. m  e.  a  m 
 ~<  a ) ) )   &    |-  M  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )   &    |-  J  = OrdIso ( Q ,  ( a  X.  a ) )   =>    |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
 
Theoreminfxpen 7575 Every infinite ordinal is equinumerous to its cross product. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation  R is a well-ordering of  ( On  X.  On ) with the additional property that  R-initial segments of  ( x  X.  x ) (where  x is a limit ordinal) are of cardinality at most  x. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
 
Theoremxpomen 7576 The cross product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( om  X.  om )  ~~  om
 
Theoreminfxpidm2 7577 The cross product of an infinite set with itself is idempotent. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 8117. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A ) 
 ->  ( A  X.  A )  ~~  A )
 
Theoreminfxpenc 7578* A canonical version of infxpen 7575, by a completely different approach (although it uses infxpen 7575 via xpomen 7576). Using Cantor's normal form, we can show that  A  ^o  B respects equinumerosity (oef1o 7334), so that all the steps of  ( om ^ W
)  x.  ( om
^ W )  ~~  om
^ ( 2 W )  ~~  ( om ^
2 ) ^ W  ~~  om ^ W can be verified using bijections to do the ordinal commutations. (The assumption on  N can be satisfied using cnfcom3c 7342.) (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  om  C_  A )   &    |-  ( ph  ->  W  e.  ( On  \  1o ) )   &    |-  ( ph  ->  F : ( om  ^o  2o ) -1-1-onto-> om )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   &    |-  ( ph  ->  N : A -1-1-onto-> ( om  ^o  W ) )   &    |-  K  =  ( y  e.  { x  e.  ( ( om  ^o  2o )  ^m  W )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( F  o.  (
 y  o.  `' (  _I  |`  W ) ) ) )   &    |-  H  =  ( ( ( om CNF  W )  o.  K )  o.  `' ( ( om  ^o  2o ) CNF  W )
 )   &    |-  L  =  ( y  e.  { x  e.  ( om  ^m  ( W  .o  2o ) )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( (  _I  |`  om )  o.  ( y  o.  `' ( Y  o.  `' X ) ) ) )   &    |-  X  =  ( z  e.  2o ,  w  e.  W  |->  ( ( W  .o  z )  +o  w ) )   &    |-  Y  =  ( z  e.  2o ,  w  e.  W  |->  ( ( 2o  .o  w )  +o  z
 ) )   &    |-  J  =  ( ( ( om CNF  ( 2o  .o  W ) )  o.  L )  o.  `' ( om CNF  ( W  .o  2o ) ) )   &    |-  Z  =  ( x  e.  ( om  ^o  W ) ,  y  e.  ( om  ^o  W ) 
 |->  ( ( ( om  ^o  W )  .o  x )  +o  y ) )   &    |-  T  =  ( x  e.  A ,  y  e.  A  |->  <. ( N `  x ) ,  ( N `  y ) >. )   &    |-  G  =  ( `' N  o.  ( ( ( H  o.  J )  o.  Z )  o.  T ) )   =>    |-  ( ph  ->  G : ( A  X.  A ) -1-1-onto-> A )
 
Theoreminfxpenc2lem1 7579* Lemma for infxpenc2 7582. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( n `
  b ) : b -1-1-onto-> ( om  ^o  w ) ) )   &    |-  W  =  ( `' ( x  e.  ( On  \  1o )  |->  ( om  ^o  x ) ) `  ran  (  n `  b
 ) )   =>    |-  ( ( ph  /\  (
 b  e.  A  /\  om  C_  b ) )  ->  ( W  e.  ( On  \  1o )  /\  ( n `  b ) : b -1-1-onto-> ( om  ^o  W ) ) )
 
Theoreminfxpenc2lem2 7580* Lemma for infxpenc2 7582. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( n `
  b ) : b -1-1-onto-> ( om  ^o  w ) ) )   &    |-  W  =  ( `' ( x  e.  ( On  \  1o )  |->  ( om  ^o  x ) ) `  ran  (  n `  b
 ) )   &    |-  ( ph  ->  F : ( om  ^o  2o ) -1-1-onto-> om )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   &    |-  K  =  ( y  e.  { x  e.  ( ( om  ^o  2o )  ^m  W )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( F  o.  (
 y  o.  `' (  _I  |`  W ) ) ) )   &    |-  H  =  ( ( ( om CNF  W )  o.  K )  o.  `' ( ( om  ^o  2o ) CNF  W )
 )   &    |-  L  =  ( y  e.  { x  e.  ( om  ^m  ( W  .o  2o ) )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( (  _I  |`  om )  o.  ( y  o.  `' ( Y  o.  `' X ) ) ) )   &    |-  X  =  ( z  e.  2o ,  w  e.  W  |->  ( ( W  .o  z )  +o  w ) )   &    |-  Y  =  ( z  e.  2o ,  w  e.  W  |->  ( ( 2o  .o  w )  +o  z
 ) )   &    |-  J  =  ( ( ( om CNF  ( 2o  .o  W ) )  o.  L )  o.  `' ( om CNF  ( W  .o  2o ) ) )   &    |-  Z  =  ( x  e.  ( om  ^o  W ) ,  y  e.  ( om  ^o  W ) 
 |->  ( ( ( om  ^o  W )  .o  x )  +o  y ) )   &    |-  T  =  ( x  e.  b ,  y  e.  b  |->  <. ( ( n `
  b ) `  x ) ,  (
 ( n `  b
 ) `  y ) >. )   &    |-  G  =  ( `' ( n `  b
 )  o.  ( ( ( H  o.  J )  o.  Z )  o.  T ) )   =>    |-  ( ph  ->  E. g A. b  e.  A  ( om  C_  b  ->  ( g `  b
 ) : ( b  X.  b ) -1-1-onto-> b ) )
 
Theoreminfxpenc2lem3 7581* Lemma for infxpenc2 7582. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( n `
  b ) : b -1-1-onto-> ( om  ^o  w ) ) )   &    |-  W  =  ( `' ( x  e.  ( On  \  1o )  |->  ( om  ^o  x ) ) `  ran  (  n `  b
 ) )   &    |-  ( ph  ->  F : ( om  ^o  2o ) -1-1-onto-> om )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   =>    |-  ( ph  ->  E. g A. b  e.  A  ( om  C_  b  ->  ( g `  b ) : ( b  X.  b ) -1-1-onto-> b ) )
 
Theoreminfxpenc2 7582* Existence form of infxpenc 7578. A "uniform" or "canonical" version of infxpen 7575, asserting the existence of a single function  g that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( A  e.  On  ->  E. g A. b  e.  A  ( om  C_  b  ->  ( g `  b
 ) : ( b  X.  b ) -1-1-onto-> b ) )
 
Theoremiunmapdisj 7583* The union  U_ n  e.  C ( A  ^m  n ) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.)
 |- 
 E* n ( n  e.  C  /\  B  e.  ( A  ^m  n ) )
 
Theoremfseqenlem1 7584* Lemma for fseqen 7587. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  F : ( A  X.  A ) -1-1-onto-> A )   &    |-  G  = seq𝜔 ( ( n  e. 
 _V ,  f  e. 
 _V  |->  ( x  e.  ( A  ^m  suc  n )  |->  ( ( f `
  ( x  |`  n ) ) F ( x `  n ) ) ) ) ,  { <. (/) ,  B >. } )   =>    |-  ( ( ph  /\  C  e.  om )  ->  ( G `  C ) : ( A  ^m  C ) -1-1-> A )
 
Theoremfseqenlem2 7585* Lemma for fseqen 7587. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  F : ( A  X.  A ) -1-1-onto-> A )   &    |-  G  = seq𝜔 ( ( n  e. 
 _V ,  f  e. 
 _V  |->  ( x  e.  ( A  ^m  suc  n )  |->  ( ( f `
  ( x  |`  n ) ) F ( x `  n ) ) ) ) ,  { <. (/) ,  B >. } )   &    |-  K  =  ( y  e.  U_ k  e.  om  ( A  ^m  k )  |->  <. dom  y ,  ( ( G `  dom  y ) `  y
 ) >. )   =>    |-  ( ph  ->  K : U_ k  e.  om  ( A  ^m  k )
 -1-1-> ( om  X.  A ) )
 
Theoremfseqdom 7586* One half of fseqen 7587. (Contributed by Mario Carneiro, 18-Nov-2014.)
 |-  ( A  e.  V  ->  ( om  X.  A ) 
 ~<_  U_ n  e.  om  ( A  ^m  n ) )
 
Theoremfseqen 7587* A set that is equinumerous to its cross product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.)
 |-  ( ( ( A  X.  A )  ~~  A  /\  A  =/=  (/) )  ->  U_ n  e.  om  ( A  ^m  n )  ~~  ( om  X.  A ) )
 
Theoreminfpwfidom 7588 The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption 
( ~P A  i^i  Fin )  e.  _V because this theorem also implies that  A is a set if  ~P A  i^i  Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
 
Theoremdfac8alem 7589* Lemma for dfac8a 7590. If the power set of a set has a choice function, then the set is numerable. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)
 |-  F  = recs ( G )   &    |-  G  =  ( f  e.  _V  |->  ( g `  ( A 
 \  ran  f )
 ) )   =>    |-  ( A  e.  C  ->  ( E. g A. y  e.  ~P  A ( y  =/=  (/)  ->  (
 g `  y )  e.  y )  ->  A  e.  dom  card ) )
 
Theoremdfac8a 7590* Numeration theorem: every set with a choice function on its power set is numerable. With AC, this reduces to the statement that every set is numerable. Similar to Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)
 |-  ( A  e.  B  ->  ( E. h A. y  e.  ~P  A ( y  =/=  (/)  ->  ( h `  y )  e.  y )  ->  A  e.  dom  card ) )
 
Theoremdfac8b 7591* The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  dom  card 
 ->  E. x  x  We  A )
 
Theoremdfac8clem 7592* Lemma for dfac8c 7593. (Contributed by Mario Carneiro, 10-Jan-2013.)
 |-  F  =  ( s  e.  ( A  \  { (/) } )  |->  (
 iota_ a  e.  s A. b  e.  s  -.  b r a ) )   =>    |-  ( A  e.  B  ->  ( E. r  r  We  U. A  ->  E. f A. z  e.  A  ( z  =/=  (/)  ->  ( f `  z )  e.  z
 ) ) )
 
Theoremdfac8c 7593* If the union of a set is well-orderable, then the set has a choice function. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( A  e.  B  ->  ( E. r  r  We  U. A  ->  E. f A. z  e.  A  ( z  =/=  (/)  ->  ( f `  z )  e.  z
 ) ) )
 
Theoremac10ct 7594* A proof of the Well ordering theorem weth 8055, an Axiom of Choice equivalent, restricted to sets dominated by some ordinal (in particular finite sets and countable sets), proven in ZF without AC. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( E. y  e. 
 On  A  ~<_  y  ->  E. x  x  We  A )
 
Theoremween 7595* A set is numerable iff it can be well ordered. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( A  e.  dom  card  <->  E. r  r  We  A )
 
Theoremac5num 7596* A version of ac5b 8038 with the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( U. A  e.  dom  card  /\  -.  (/)  e.  A )  ->  E. f ( f : A --> U. A  /\  A. x  e.  A  ( f `  x )  e.  x )
 )
 
Theoremondomen 7597 If a set is dominated by an ordinal, then it is numerable. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( ( A  e.  On  /\  B  ~<_  A ) 
 ->  B  e.  dom  card )
 
Theoremnumdom 7598 A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  ~<_  A ) 
 ->  B  e.  dom  card )
 
Theoremssnum 7599 A subset of a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  C_  A )  ->  B  e.  dom  card
 )
 
Theoremonssnum 7600 All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.)
 |-  ( ( A  e.  V  /\  A  C_  On )  ->  A  e.  dom  card
 )
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