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Theorem List for Metamath Proof Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcardalephex 7601* Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse. (Contributed by NM, 5-Nov-2003.)
 |-  ( om  C_  A  ->  ( ( card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
 
Theoreminfenaleph 7602* An infinite numerable set is equinumerous to an infinite initial ordinal. (Contributed by Jeff Hankins, 23-Oct-2009.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A ) 
 ->  E. x  e.  ran  aleph  x  ~~  A )
 
Theoremisinfcard 7603 Two ways to express the property of being a transfinite cardinal. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph
 )
 
Theoremiscard3 7604 Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( card `  A )  =  A  <->  A  e.  ( om  u.  ran  aleph ) )
 
Theoremcardnum 7605 Two ways to express the class of all cardinal numbers, which consists of the finite ordinals in  om plus the transfinite alephs. (Contributed by NM, 10-Sep-2004.)
 |- 
 { x  |  (
 card `  x )  =  x }  =  ( om  u.  ran  aleph )
 
Theoremalephinit 7606* An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  e.  ran  aleph  <->  A. x  e.  On  ( A  ~<_  x  ->  A 
 C_  x ) ) )
 
Theoremcarduniima 7607 The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
 |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  ->  U. ( F " A )  e.  ( om  u.  ran  aleph ) ) )
 
Theoremcardinfima 7608* If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
 |-  ( A  e.  B  ->  ( ( F : A
 --> ( om  u.  ran  aleph
 )  /\  E. x  e.  A  ( F `  x )  e.  ran  aleph
 )  ->  U. ( F
 " A )  e. 
 ran  aleph ) )
 
Theoremalephiso 7609 Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90. (Contributed by NM, 3-Aug-2004.)
 |-  aleph 
 Isom  _E  ,  _E  ( On ,  { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) } )
 
Theoremalephprc 7610 The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of [TakeutiZaring] p. 90. (Contributed by NM, 11-Nov-2003.)
 |- 
 -.  ran  aleph  e.  _V
 
Theoremalephsson 7611 The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
 |- 
 ran  aleph  C_  On
 
Theoremunialeph 7612 The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
 |- 
 U. ran  aleph  =  On
 
Theoremalephsmo 7613 The aleph function is strictly monotone. (Contributed by by Mario Carneiro, 15-Mar-2013.)
 |- 
 Smo  aleph
 
Theoremalephf1ALT 7614 The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. (Contributed by by Mario Carneiro, 15-Mar-2013.) (Proof modification is discouraged.)
 |-  aleph : On -1-1-> On
 
Theoremalephfplem1 7615 Lemma for alephfp 7619. (Contributed by NM, 6-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( H `  (/) )  e. 
 ran  aleph
 
Theoremalephfplem2 7616* Lemma for alephfp 7619. (Contributed by NM, 6-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( w  e.  om  ->  ( H `  suc  w )  =  ( aleph `  ( H `  w ) ) )
 
Theoremalephfplem3 7617* Lemma for alephfp 7619. (Contributed by NM, 6-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( v  e.  om  ->  ( H `  v
 )  e.  ran  aleph )
 
Theoremalephfplem4 7618 Lemma for alephfp 7619. (Contributed by NM, 5-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |- 
 U. ( H " om )  e.  ran  aleph
 
Theoremalephfp 7619 The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 7620 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( aleph `  U. ( H
 " om ) )  =  U. ( H
 " om )
 
Theoremalephfp2 7620 The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 7619 for an actual example of a fixed point. Compare the inequality alephle 7599 that holds in general. Note that if  x is a fixed point, then  aleph `  aleph `  aleph ` ...  aleph `  x  =  x. (Contributed by NM, 6-Nov-2004.) (Revised by Mario Carneiro, 15-May-2015.)
 |- 
 E. x  e.  On  ( aleph `  x )  =  x
 
Theoremalephval3 7621* An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.)
 |-  ( A  e.  On  ->  ( aleph `  A )  =  |^| { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y )
 ) } )
 
Theoremalephsucpw2 7622 The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8182 or gchaleph2 8178.) The transposed form alephsucpw 8072 cannot be proven without the AC, and is in fact equlvalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.)
 |- 
 -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
 
Theoremmappwen 7623 Power rule for cardinal arithmetic. Theorem 11.21 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A 
 ^m  B )  ~~  ~P B )
 
Theoremfinnisoeu 7624* A finite totally ordered set has a unique order isomorphism to a finite ordinal. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 26-Jun-2015.)
 |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E! f  f 
 Isom  _E  ,  R  ( ( card `  A ) ,  A ) )
 
Theoremiunfictbso 7625 Countability of a countable union of finite sets with an strict (not globally well) order fulfilling the choice role. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( ( A  ~<_  om  /\  A  C_  Fin  /\  B  Or  U. A )  ->  U. A  ~<_  om )
 
2.6.8  Axiom of Choice equivalents
 
Syntaxwac 7626 Wff for an abbreviation of the axiom of choice.
 wff CHOICE
 
Definitiondf-ac 7627* The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There is a slight problem with taking the exact form of ax-ac 7969 as our definition, because the equivalence to more standard forms (dfac2 7641) requires the Axiom of Regularity, which we often try to avoid. Thus we take the first of the "textbook forms" as the definition and derive the form of ax-ac 7969 itself as dfac0 7643. (Contributed by Mario Carneiro, 22-Feb-2015.)

 |-  (CHOICE  <->  A. x E. f ( f  C_  x  /\  f  Fn  dom  x )
 )
 
Theoremaceq1 7628* Equivalence of two versions of the Axiom of Choice ax-ac 7969. The proof uses neither AC nor the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  (
 z  e.  u  /\  v  e.  u )  <->  E. y A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. x A. z ( E. x ( ( z  e.  w  /\  w  e.  x )  /\  (
 z  e.  x  /\  x  e.  y )
 ) 
 <->  z  =  x ) ) )
 
Theoremaceq0 7629* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 7969. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  (
 z  e.  u  /\  v  e.  u )  <->  E. y A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. v A. u ( E. t
 ( ( u  e.  w  /\  w  e.  t )  /\  ( u  e.  t  /\  t  e.  y )
 ) 
 <->  u  =  v ) ) )
 
Theoremaceq2 7630* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  (
 z  e.  u  /\  v  e.  u )  <->  E. y A. z  e.  x  ( z  =/=  (/)  ->  E! w  e.  z  E. v  e.  y  ( z  e.  v  /\  w  e.  v ) ) )
 
Theoremaceq3lem 7631* Lemma for dfac3 7632. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  F  =  ( w  e.  dom  y  |->  ( f `  { u  |  w y u }
 ) )   =>    |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  (
 f `  z )  e.  z )  ->  E. f
 ( f  C_  y  /\  f  Fn  dom  y
 ) )
 
Theoremdfac3 7632* Equivalence of two versions of the Axiom of Choice. The left-hand side is is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Stefan O'Rear, 22-Feb-2015.)
 |-  (CHOICE  <->  A. x E. f A. z  e.  x  (
 z  =/=  (/)  ->  (
 f `  z )  e.  z ) )
 
Theoremdfac4 7633* Equivalence of two versions of the Axiom of Choice. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (CHOICE  <->  A. x E. f ( f  Fn  x  /\  A. z  e.  x  ( z  =/=  (/)  ->  (
 f `  z )  e.  z ) ) )
 
Theoremdfac5lem1 7634* Lemma for dfac5 7639. (Contributed by NM, 12-Apr-2004.)
 |-  ( E! v  v  e.  ( ( { w }  X.  w )  i^i  y )  <->  E! g ( g  e.  w  /\  <. w ,  g >.  e.  y
 ) )
 
Theoremdfac5lem2 7635* Lemma for dfac5 7639. (Contributed by NM, 12-Apr-2004.)
 |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }   =>    |-  ( <. w ,  g >.  e. 
 U. A  <->  ( w  e.  h  /\  g  e.  w ) )
 
Theoremdfac5lem3 7636* Lemma for dfac5 7639. (Contributed by NM, 12-Apr-2004.)
 |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }   =>    |-  (
 ( { w }  X.  w )  e.  A  <->  ( w  =/=  (/)  /\  w  e.  h ) )
 
Theoremdfac5lem4 7637* Lemma for dfac5 7639. (Contributed by NM, 11-Apr-2004.)
 |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }   &    |-  B  =  ( U. A  i^i  y )   &    |-  ( ph  <->  A. x ( (
 A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/= 
 w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) ) )   =>    |-  ( ph  ->  E. y A. z  e.  A  E! v  v  e.  ( z  i^i  y ) )
 
Theoremdfac5lem5 7638* Lemma for dfac5 7639. (Contributed by NM, 12-Apr-2004.)
 |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }   &    |-  B  =  ( U. A  i^i  y )   &    |-  ( ph  <->  A. x ( (
 A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/= 
 w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) ) )   =>    |-  ( ph  ->  E. f A. w  e.  h  ( w  =/=  (/)  ->  (
 f `  w )  e.  w ) )
 
Theoremdfac5 7639* Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x ( ( A. z  e.  x  z  =/= 
 (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/= 
 w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) ) )
 
Theoremdfac2a 7640* Our Axiom of Choice (in the form of ac3 7972) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 7641 for the converse (which does use the Axiom of Regularity). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  ( A. x E. y A. z  e.  x  ( z  =/=  (/)  ->  E! w  e.  z  E. v  e.  y  (
 z  e.  v  /\  w  e.  v )
 )  -> CHOICE )
 
Theoremdfac2 7641* Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 7972). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 7195 and preleq 7202 that are referenced in the proof each make use of Regularity for their derivations. (The reverse implication can be derived without using Regularity; see dfac2a 7640.) TODO: Fix label in comment, and put label changes into list at top of set.mm. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (CHOICE  <->  A. x E. y A. z  e.  x  (
 z  =/=  (/)  ->  E! w  e.  z  E. v  e.  y  (
 z  e.  v  /\  w  e.  v )
 ) )
 
Theoremdfac7 7642* Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 7971). The proof does not depend AC on but does depend on the Axiom of Regularity. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  (
 z  e.  u  /\  v  e.  u )
 )
 
Theoremdfac0 7643* Equivalence of two versions of the Axiom of Choice. The proof uses the Axiom of Regularity. The right-hand side is our original ax-ac 7969. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x E. y A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. v A. u ( E. t ( ( u  e.  w  /\  w  e.  t )  /\  ( u  e.  t  /\  t  e.  y
 ) )  <->  u  =  v
 ) ) )
 
Theoremdfac1 7644* Equivalence of two versions of the Axiom of Choice ax-ac 7969. The proof uses the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x E. y A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. x A. z
 ( E. x ( ( z  e.  w  /\  w  e.  x )  /\  ( z  e.  x  /\  x  e.  y ) )  <->  z  =  x ) ) )
 
Theoremdfac8 7645* A proof of the equivalency of the Well Ordering Theorem weth 8006 and the Axiom of Choice ac7 7984. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  (CHOICE  <->  A. x E. r  r  We  x )
 
Theoremdfac9 7646* Equivalence of the axiom of choice with a statement related to ac9 7994; definition AC3 of [Schechter] p. 139. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (CHOICE  <->  A. f ( ( Fun  f  /\  (/)  e/  ran  f )  ->  X_ x  e.  dom  f ( f `
  x )  =/=  (/) ) )
 
Theoremdfac10 7647 Axiom of Choice equivalent: the cardinality function measures every set. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  (CHOICE  <->  dom  card  =  _V )
 
Theoremdfac10c 7648* Axiom of Choice equivalent: every set is equinumerous to an ordinal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  (CHOICE  <->  A. x E. y  e. 
 On  y  ~~  x )
 
Theoremdfac10b 7649 Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 7627). (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  (CHOICE  <->  ( 
 ~~  " On )  =  _V )
 
Theoremacacni 7650 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( (CHOICE 
 /\  A  e.  V )  -> AC  A  =  _V )
 
Theoremdfacacn 7651 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  (CHOICE  <->  A. xAC  x  =  _V )
 
Theoremdfac13 7652 The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  (CHOICE  <->  A. x  x  e. AC  x )
 
Theoremdfac12lem1 7653* Lemma for dfac12 7659. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  F : ~P (har `  ( R1 `  A ) ) -1-1-> On )   &    |-  G  = recs (
 ( x  e.  _V  |->  ( y  e.  ( R1 `  dom  x ) 
 |->  if ( dom  x  =  U. dom  x ,  ( ( suc  U. ran  U.
 ran  x  .o  ( rank `  y ) )  +o  ( ( x `
  suc  ( rank `  y ) ) `  y ) ) ,  ( F `  (
 ( `'OrdIso (  _E  ,  ran  (  x `  U. dom  x ) )  o.  ( x `  U. dom  x ) ) " y
 ) ) ) ) ) )   &    |-  ( ph  ->  C  e.  On )   &    |-  H  =  ( `'OrdIso (  _E  ,  ran  (  G `  U. C ) )  o.  ( G `  U. C ) )   =>    |-  ( ph  ->  ( G `  C )  =  ( y  e.  ( R1 `  C )  |->  if ( C  =  U. C ,  ( ( suc  U. ran  U. ( G " C )  .o  ( rank `  y )
 )  +o  ( ( G `  suc  ( rank `  y ) ) `  y ) ) ,  ( F `  ( H " y ) ) ) ) )
 
Theoremdfac12lem2 7654* Lemma for dfac12 7659. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  F : ~P (har `  ( R1 `  A ) ) -1-1-> On )   &    |-  G  = recs (
 ( x  e.  _V  |->  ( y  e.  ( R1 `  dom  x ) 
 |->  if ( dom  x  =  U. dom  x ,  ( ( suc  U. ran  U.
 ran  x  .o  ( rank `  y ) )  +o  ( ( x `
  suc  ( rank `  y ) ) `  y ) ) ,  ( F `  (
 ( `'OrdIso (  _E  ,  ran  (  x `  U. dom  x ) )  o.  ( x `  U. dom  x ) ) " y
 ) ) ) ) ) )   &    |-  ( ph  ->  C  e.  On )   &    |-  H  =  ( `'OrdIso (  _E  ,  ran  (  G `  U. C ) )  o.  ( G `  U. C ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  A. z  e.  C  ( G `  z ) : ( R1 `  z ) -1-1-> On )   =>    |-  ( ph  ->  ( G `  C ) : ( R1 `  C )
 -1-1-> On )
 
Theoremdfac12lem3 7655* Lemma for dfac12 7659. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  F : ~P (har `  ( R1 `  A ) ) -1-1-> On )   &    |-  G  = recs (
 ( x  e.  _V  |->  ( y  e.  ( R1 `  dom  x ) 
 |->  if ( dom  x  =  U. dom  x ,  ( ( suc  U. ran  U.
 ran  x  .o  ( rank `  y ) )  +o  ( ( x `
  suc  ( rank `  y ) ) `  y ) ) ,  ( F `  (
 ( `'OrdIso (  _E  ,  ran  (  x `  U. dom  x ) )  o.  ( x `  U. dom  x ) ) " y
 ) ) ) ) ) )   =>    |-  ( ph  ->  ( R1 `  A )  e. 
 dom  card )
 
Theoremdfac12r 7656 The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 7659 does not assume the Axiom of Regularity. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( A. x  e. 
 On  ~P x  e.  dom  card  <->  U. ( R1 " On )  C_  dom  card )
 
Theoremdfac12k 7657* Equivalence of dfac12 7659 and dfac12a 7658, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  ( A. x  e. 
 On  ~P x  e.  dom  card  <->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
 
Theoremdfac12a 7658 The axiom of choice holds iff every ordinal has a well-orderable powerset. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  (CHOICE  <->  A. x  e.  On  ~P x  e.  dom  card )
 
Theoremdfac12 7659 The axiom of choice holds iff every aleph has a well-orderable powerset. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  (CHOICE  <->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
 
Theoremkmlem1 7660* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2. (Contributed by NM, 5-Apr-2004.)
 |-  ( A. x ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ph )  ->  E. y A. z  e.  x  ps )  ->  A. x ( A. z  e.  x  A. w  e.  x  ph  ->  E. y A. z  e.  x  ( z  =/=  (/)  ->  ps ) ) )
 
Theoremkmlem2 7661* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
 |-  ( E. y A. z  e.  x  ( ph  ->  E! w  w  e.  ( z  i^i  y ) )  <->  E. y ( -.  y  e.  x  /\  A. z  e.  x  (
 ph  ->  E! w  w  e.  ( z  i^i  y ) ) ) )
 
Theoremkmlem3 7662* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. The right-hand side is part of the hypothesis of 4. (Contributed by NM, 25-Mar-2004.)
 |-  ( ( z  \  U. ( x  \  {
 z } ) )  =/=  (/)  <->  E. v  e.  z  A. w  e.  x  ( z  =/=  w  ->  -.  v  e.  (
 z  i^i  w )
 ) )
 
Theoremkmlem4 7663* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
 |-  ( ( w  e.  x  /\  z  =/= 
 w )  ->  (
 ( z  \  U. ( x  \  { z } ) )  i^i 
 w )  =  (/) )
 
Theoremkmlem5 7664* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
 |-  ( ( w  e.  x  /\  z  =/= 
 w )  ->  (
 ( z  \  U. ( x  \  { z } ) )  i^i  ( w  \  U. ( x  \  { w } ) ) )  =  (/) )
 
Theoremkmlem6 7665* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
 |-  ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  (
 ph  ->  A  =  (/) ) )  ->  A. z  e.  x  E. v  e.  z  A. w  e.  x  ( ph  ->  -.  v  e.  A ) )
 
Theoremkmlem7 7666* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
 |-  ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) )  ->  -.  E. z  e.  x  A. v  e.  z  E. w  e.  x  (
 z  =/=  w  /\  v  e.  ( z  i^i  w ) ) )
 
Theoremkmlem8 7667* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 4-Apr-2004.)
 |-  ( ( -.  E. z  e.  u  A. w  e.  z  ps  ->  E. y A. z  e.  u  ( z  =/= 
 (/)  ->  E! w  w  e.  ( z  i^i  y ) ) )  <-> 
 ( E. z  e.  u  A. w  e.  z  ps  \/  E. y ( -.  y  e.  u  /\  A. z  e.  u  E! w  w  e.  ( z  i^i  y ) ) ) )
 
Theoremkmlem9 7668* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |- 
 A. z  e.  A  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )
 
Theoremkmlem10 7669* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |-  ( A. h (
 A. z  e.  h  A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )  ->  E. y A. z  e.  h  ph )  ->  E. y A. z  e.  A  ph )
 
Theoremkmlem11 7670* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |-  ( z  e.  x  ->  ( z  i^i  U. A )  =  (
 z  \  U. ( x 
 \  { z }
 ) ) )
 
Theoremkmlem12 7671* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |-  ( A. z  e.  x  ( z  \  U. ( x  \  {
 z } ) )  =/=  (/)  ->  ( A. z  e.  A  (
 z  =/=  (/)  ->  E! v  v  e.  (
 z  i^i  y )
 )  ->  A. z  e.  x  ( z  =/=  (/)  ->  E! v  v  e.  ( z  i^i  ( y  i^i  U. A ) ) ) ) )
 
Theoremkmlem13 7672* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 5-Apr-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |-  ( A. x ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  (
 z  =/=  w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )  <->  A. x ( -. 
 E. z  e.  x  A. v  e.  z  E. w  e.  x  (
 z  =/=  w  /\  v  e.  ( z  i^i  w ) )  ->  E. y A. z  e.  x  ( z  =/=  (/)  ->  E! v  v  e.  ( z  i^i  y ) ) ) )
 
Theoremkmlem14 7673* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
 |-  ( ph  <->  ( z  e.  y  ->  ( (
 v  e.  x  /\  y  =/=  v )  /\  z  e.  v )
 ) )   &    |-  ( ps  <->  ( z  e.  x  ->  ( (
 v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) )   &    |-  ( ch  <->  A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )   =>    |-  ( E. z  e.  x  A. v  e.  z  E. w  e.  x  ( z  =/= 
 w  /\  v  e.  ( z  i^i  w ) )  <->  E. y A. z E. v A. u ( y  e.  x  /\  ph ) )
 
Theoremkmlem15 7674* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
 |-  ( ph  <->  ( z  e.  y  ->  ( (
 v  e.  x  /\  y  =/=  v )  /\  z  e.  v )
 ) )   &    |-  ( ps  <->  ( z  e.  x  ->  ( (
 v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) )   &    |-  ( ch  <->  A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )   =>    |-  ( ( -.  y  e.  x  /\  ch )  <->  A. z E. v A. u ( -.  y  e.  x  /\  ps )
 )
 
Theoremkmlem16 7675* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
 |-  ( ph  <->  ( z  e.  y  ->  ( (
 v  e.  x  /\  y  =/=  v )  /\  z  e.  v )
 ) )   &    |-  ( ps  <->  ( z  e.  x  ->  ( (
 v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) )   &    |-  ( ch  <->  A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )   =>    |-  ( ( E. z  e.  x  A. v  e.  z  E. w  e.  x  ( z  =/= 
 w  /\  v  e.  ( z  i^i  w ) )  \/  E. y
 ( -.  y  e.  x  /\  ch )
 ) 
 <-> 
 E. y A. z E. v A. u ( ( y  e.  x  /\  ph )  \/  ( -.  y  e.  x  /\  ps ) ) )
 
Theoremdfackm 7676* Equivalence of the Axiom of Choice and Maes' AC ackm 7976. The proof consists of lemmas kmlem1 7660 through kmlem16 7675 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 7639 with biid 229) establishes the AC equivalence shown by Maes' writeup. The left-hand-side AC shown here was chosen because it is shorter to display. (Contributed by NM, 13-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x E. y A. z E. v A. u ( ( y  e.  x  /\  ( z  e.  y  ->  (
 ( v  e.  x  /\  -.  y  =  v )  /\  z  e.  v ) ) )  \/  ( -.  y  e.  x  /\  ( z  e.  x  ->  (
 ( v  e.  z  /\  v  e.  y
 )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) ) ) )
 
2.6.9  Cardinal number arithmetic
 
Syntaxccda 7677 Extend class definition to include cardinal number addition.
 class  +c
 
Definitiondf-cda 7678* Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 7680 for its value and a description. (Contributed by NM, 24-Sep-2004.)
 |- 
 +c  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
 
Theoremcdafn 7679 Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 +c  Fn  ( _V  X. 
 _V )
 
Theoremcdaval 7680 Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while cross product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 8055, carddom 8058, and cardsdom 8059. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +c  B )  =  (
 ( A  X.  { (/)
 } )  u.  ( B  X.  { 1o }
 ) ) )
 
Theoremuncdadom 7681 Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B )  ~<_  ( A  +c  B ) )
 
Theoremcdaun 7682 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by NM, 5-Apr-2007.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B ) 
 ~~  ( A  u.  B ) )
 
Theoremcdaen 7683 Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  +c  C )  ~~  ( B  +c  D ) )
 
Theoremcdaenun 7684 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A  +c  C ) 
 ~~  ( B  u.  D ) )
 
Theoremcda1en 7685 Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  -.  A  e.  A )  ->  ( A  +c  1o )  ~~  suc 
 A )
 
Theoremcda1dif 7686 Adding and subtracting one gives back the original set. Similar to pncan 8937 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o )  \  { B } )  ~~  A )
 
Theorempm110.643 7687 1+1=2 for cardinal number addition, derived from pm54.43 7517 as promised. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 7449), but after applying definitions, our theorem is equivalent. The comment for cdaval 7680 explains why we use  ~~ instead of =. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
 |-  ( 1o  +c  1o )  ~~  2o
 
Theorempm110.643ALT 7688 Alternate proof of pm110.643 7687. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.)
 |-  ( 1o  +c  1o )  ~~  2o
 
Theoremcda0en 7689 Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  V  ->  ( A  +c  (/) )  ~~  A )
 
Theoremxp2cda 7690 Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  V  ->  ( A  X.  2o )  =  ( A  +c  A ) )
 
Theoremcdacomen 7691 Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  +c  B )  ~~  ( B  +c  A )
 
Theoremcdaassen 7692 Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( ( A  +c  B )  +c  C ) 
 ~~  ( A  +c  ( B  +c  C ) ) )
 
Theoremxpcdaen 7693 Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( A  X.  ( B  +c  C ) ) 
 ~~  ( ( A  X.  B )  +c  ( A  X.  C ) ) )
 
Theoremmapcdaen 7694 Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( A  ^m  ( B  +c  C ) ) 
 ~~  ( ( A 
 ^m  B )  X.  ( A  ^m  C ) ) )
 
Theorempwcdaen 7695 Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ( A  +c  B )  ~~  ( ~P A  X.  ~P B ) )
 
Theoremcdadom1 7696 Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  ~<_  B  ->  ( A  +c  C )  ~<_  ( B  +c  C ) )
 
Theoremcdadom2 7697 Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  ~<_  B  ->  ( C  +c  A )  ~<_  ( C  +c  B ) )
 
Theoremcdadom3 7698 A set is dominated by its cardinal sum with another. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  ( A  +c  B ) )
 
Theoremcdaxpdom 7699 Cross product dominates disjoint union for sets with cardinality greater than 1. Similar to Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( ( 1o  ~<  A 
 /\  1o  ~<  B ) 
 ->  ( A  +c  B ) 
 ~<_  ( A  X.  B ) )
 
Theoremcdafi 7700 The cardinal sum of two finite sets is finite. (Contributed by NM, 22-Oct-2004.)
 |-  ( ( A  ~<  om 
 /\  B  ~<  om )  ->  ( A  +c  B )  ~<  om )
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