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Theorem List for Metamath Proof Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcardval3 7601* 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 7602 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 7603 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 7604* The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 8184, 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 7605 Any ordinal number is equinumerous to its cardinal number. Unlike cardid 8185, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  On  ->  ( card `  A )  ~~  A )
 
Theoremcardonle 7606 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 7607 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
 |-  ( card `  (/) )  =  (/)
 
Theoremcardidm 7608 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 7609* 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 7610 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 7611 A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)
 |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
 
Theoremcardnn 7612 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 7613 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 7614 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 7615 If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7616 are meant to replace carden 8189 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)
 |-  ( ( ( card `  A )  =  (
 card `  B )  /\  ( card `  A )  =/= 
 (/) )  ->  A  ~~  B )
 
Theoremcarden2b 7616 If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7615 are meant to replace carden 8189 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 7617* 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 7618 A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)
 |-  ( A  e.  V  ->  ( card `  { A }
 )  =  1o )
 
Theoremcarddomi2 7619 Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8192, 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 7620 A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  ( A  ~<  ( card `  B )  ->  A  ~<  B )
 
Theoremcardlim 7621 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 7622 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7623 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( A  e.  ( card `  B )  ->  A  ~<  B )
 
Theoremcardsdomel 7623 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 7624* 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 7625* 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 7626 Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8192, 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 7627 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 7628* Lemma for cardprc 7629. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  A  =  { x  |  ( card `  x )  =  x }   =>    |- 
 -.  A  e.  _V
 
Theoremcardprc 7629 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 8199 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7275 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 7630* 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 7631* 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 7632 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 7633 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 7634 The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 7633. (Contributed by NM, 7-Nov-2008.)
 |-  ( A  e.  om  ->  ( card `  suc  A )  =  suc  ( card `  A ) )
 
Theoremcardom 7635 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 7636 Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 8189, 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 7637 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 7638 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 7639 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 7640* An alternate version of the value of the cardinal number of a set. Compare cardval 8184. This theorem could be used to give us a simpler definition of  card in place of df-card 7588. 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 7641* An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7092 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 7642 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 7643 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 7644 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 7645* 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 7646* 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 7647* 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 7648* 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 7617), 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 7649. (Contributed by NM, 4-Nov-2013.)
 |-  ( A  ~~  1o  <->  A  e.  { x  |  (
 card `  x )  =  1o } )
 
Theorempm54.43 7649 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 7617), 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 7648. 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 7819 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 7650 Lemma for pr2ne 7651. (Contributed by FL, 17-Aug-2008.)
 |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B ) 
 ->  { A ,  B }  ~~  2o )
 
Theorempr2ne 7651 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 7652 An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { A ,  B }  ~<_  2o )
 
Theoremen2eqpr 7653 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 7654 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 7655* Lexicographical order is a well-ordering of  On 
X.  On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7656, 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 7656* 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 7657* Lemma for infxpen 7658. (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 7658 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 7659 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 7660 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 8200. (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 7661* A canonical version of infxpen 7658, by a completely different approach (although it uses infxpen 7658 via xpomen 7659). Using Cantor's normal form, we can show that  A  ^o  B respects equinumerosity (oef1o 7417), 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 7425.) (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 7662* Lemma for infxpenc2 7665. (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 7663* Lemma for infxpenc2 7665. (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 7664* Lemma for infxpenc2 7665. (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 7665* Existence form of infxpenc 7661. A "uniform" or "canonical" version of infxpen 7658, 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 7666* The union  U_ n  e.  C ( A  ^m  n ) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)
 |- 
 E* n  e.  C B  e.  ( A  ^m  n )
 
Theoremfseqenlem1 7667* Lemma for fseqen 7670. (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 7668* Lemma for fseqen 7670. (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 7669* One half of fseqen 7670. (Contributed by Mario Carneiro, 18-Nov-2014.)
 |-  ( A  e.  V  ->  ( om  X.  A ) 
 ~<_  U_ n  e.  om  ( A  ^m  n ) )
 
Theoremfseqen 7670* 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 7671 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 7672* Lemma for dfac8a 7673. 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 7673* 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 7674* 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 7675* Lemma for dfac8c 7676. (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 7676* 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 7677* A proof of the Well ordering theorem weth 8138, 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 7678* 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 7679* A version of ac5b 8121 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 7680 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 7681 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 7682 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 7683 All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.)
 |-  ( ( A  e.  V  /\  A  C_  On )  ->  A  e.  dom  card
 )
 
Theoremindcardi 7684* Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  T  e.  dom  card )   &    |-  (
 ( ph  /\  R  ~<_  T  /\  A. y ( S  ~<  R 
 ->  ch ) )  ->  ps )   &    |-  ( x  =  y  ->  ( ps  <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( x  =  y  ->  R  =  S )   &    |-  ( x  =  A  ->  R  =  T )   =>    |-  ( ph  ->  th )
 
Theoremacnrcl 7685 Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( X  e. AC  A  ->  A  e.  _V )
 
Theoremacneq 7686 Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  C  -> AC  A  = AC  C )
 
Theoremisacn 7687* The property of being a choice set of length  A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( X  e.  V  /\  A  e.  W )  ->  ( X  e. AC  A  <->  A. f  e.  (
 ( ~P X  \  { (/) } )  ^m  A ) E. g A. x  e.  A  ( g `  x )  e.  ( f `  x ) ) )
 
Theoremacni 7688* The property of being a choice set of length  A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( X  e. AC  A 
 /\  F : A --> ( ~P X  \  { (/)
 } ) )  ->  E. g A. x  e.  A  ( g `  x )  e.  ( F `  x ) )
 
Theoremacni2 7689* The property of being a choice set of length  A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( X  e. AC  A 
 /\  A. x  e.  A  ( B  C_  X  /\  B  =/=  (/) ) )  ->  E. g ( g : A --> X  /\  A. x  e.  A  (
 g `  x )  e.  B ) )
 
Theoremacni3 7690* The property of being a choice set of length  A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( y  =  ( g `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( X  e. AC  A  /\  A. x  e.  A  E. y  e.  X  ph )  ->  E. g ( g : A --> X  /\  A. x  e.  A  ps ) )
 
Theoremacnlem 7691* Construct a mapping satisfying the consequent of isacn 7687. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  (
 f `  x )
 )  ->  E. g A. x  e.  A  ( g `  x )  e.  ( f `  x ) )
 
Theoremnumacn 7692 A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  V  ->  ( X  e.  dom  card 
 ->  X  e. AC  A ) )
 
Theoremfinacn 7693 Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  Fin  -> AC  A  =  _V )
 
Theoremacndom 7694 A set with long choice sequences also has shorter choice sequences, where "shorter" here means the new index set is dominated by the old index set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  ~<_  B  ->  ( X  e. AC  B  ->  X  e. AC  A ) )
 
Theoremacnnum 7695 A set  X which has choice sequences on it of length 
~P X is well-orderable (and hence has choice sequences of every length). (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( X  e. AC  ~P X  <->  X  e.  dom  card )
 
Theoremacnen 7696 The class of choice sets of length 
A is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  ~~  B  -> AC  A  = AC  B )
 
Theoremacndom2 7697 A set smaller than one with choice sequences of length  A also has choice sequences of length 
A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( X  ~<_  Y  ->  ( Y  e. AC  A  ->  X  e. AC  A ) )
 
Theoremacnen2 7698 The class of sets with choice sequences of length  A is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( X  ~~  Y  ->  ( X  e. AC  A  <->  Y  e. AC  A ) )
 
Theoremfodomacn 7699 A version of fodom 8165 that doesn't require the Axiom of Choice ax-ac 8101. If  A has choice sequences of length  B, then any surjection from  A to  B can be inverted to an injection the other way. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e. AC  B  ->  ( F : A -onto-> B  ->  B  ~<_  A ) )
 
Theoremfodomnum 7700 A version of fodom 8165 that doesn't require the Axiom of Choice ax-ac 8101. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( A  e.  dom  card 
 ->  ( F : A -onto-> B  ->  B  ~<_  A ) )
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