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Theorem List for Metamath Proof Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfac13 7701 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 7702* Lemma for dfac12 7708. (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 7703* Lemma for dfac12 7708. (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 7704* Lemma for dfac12 7708. (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 7705 The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 7708 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 7706* Equivalence of dfac12 7708 and dfac12a 7707, 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 7707 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 7708 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 7709* 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 7710* 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 7711* 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 7712* 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 7713* 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 7714* 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 7715* 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 7716* 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 7717* 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 7718* 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 7719* 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 7720* 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 7721* 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 7722* 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 7723* 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 7724* 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 7725* Equivalence of the Axiom of Choice and Maes' AC ackm 8025. The proof consists of lemmas kmlem1 7709 through kmlem16 7724 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 7688 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 7726 Extend class definition to include cardinal number addition.
 class  +c
 
Definitiondf-cda 7727* Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 7729 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 7728 Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 +c  Fn  ( _V  X. 
 _V )
 
Theoremcdaval 7729 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 8106, carddom 8109, and cardsdom 8110. 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 7730 Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B )  ~<_  ( A  +c  B ) )
 
Theoremcdaun 7731 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 7732 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 7733 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 7734 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 7735 Adding and subtracting one gives back the original set. Similar to pncan 8990 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o )  \  { B } )  ~~  A )
 
Theorempm110.643 7736 1+1=2 for cardinal number addition, derived from pm54.43 7566 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 7498), but after applying definitions, our theorem is equivalent. The comment for cdaval 7729 explains why we use  ~~ instead of =. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
 |-  ( 1o  +c  1o )  ~~  2o
 
Theorempm110.643ALT 7737 Alternate proof of pm110.643 7736. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.)
 |-  ( 1o  +c  1o )  ~~  2o
 
Theoremcda0en 7738 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 7739 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 7740 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 7741 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 7742 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 7743 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 7744 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 7745 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 7746 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 7747 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 7748 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 7749 The cardinal sum of two finite sets is finite. (Contributed by NM, 22-Oct-2004.)
 |-  ( ( A  ~<  om 
 /\  B  ~<  om )  ->  ( A  +c  B )  ~<  om )
 
Theoremcdainflem 7750 Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.)
 |-  ( ( A  u.  B )  ~~  om  ->  ( A  ~~  om  \/  B  ~~  om ) )
 
Theoremcdainf 7751 A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( om  ~<_  A  <->  om  ~<_  ( A  +c  A ) )
 
Theoreminfcda1 7752 An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A )
 
Theorempwcda1 7753 The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  V  ->  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o )
 )
 
Theorempwcdaidm 7754 If the natural numbers inject into 
A, then  ~P A is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( om  ~<_  A  ->  ( ~P A  +c  ~P A )  ~~  ~P A )
 
Theoremcdalepw 7755 If  A is idempotent under cardinal sum and  B is dominated by the power set of  A, then so is the cardinal sum of  A and  B. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A )  ->  ( A  +c  B )  ~<_  ~P A )
 
Theoremonacda 7756 The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B )  ~~  ( A  +c  B ) )
 
Theoremcardacda 7757 The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( A  +c  B )  ~~  (
 ( card `  A )  +o  ( card `  B )
 ) )
 
Theoremcdanum 7758 The cardinal sum of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( A  +c  B )  e.  dom  card
 )
 
Theoremunnum 7759 The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( A  u.  B )  e.  dom  card
 )
 
Theoremnnacda 7760 The cardinal and ordinal sums of finite ordinals are equal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( card `  ( A  +c  B ) )  =  ( A  +o  B ) )
 
Theoremficardun 7761 The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( card `  ( A  u.  B ) )  =  ( ( card `  A )  +o  ( card `  B ) ) )
 
Theoremficardun2 7762 The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( card `  ( A  u.  B ) )  C_  ( ( card `  A )  +o  ( card `  B ) ) )
 
Theorempwsdompw 7763* Lemma for domtriom 8002. This is the equinumerosity version of the algebraic identity  sum_ k  e.  n
( 2 ^ k
)  =  ( 2 ^ n )  - 
1. (Contributed by Mario Carneiro, 7-Feb-2013.)
 |-  ( ( n  e. 
 om  /\  A. k  e. 
 suc  n ( B `
  k )  ~~  ~P k )  ->  U_ k  e.  n  ( B `  k )  ~<  ( B `
  n ) )
 
Theoremunctb 7764 The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( A  u.  B )  ~<_  om )
 
Theoreminfcdaabs 7765 Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A  /\  B 
 ~<_  A )  ->  ( A  +c  B )  ~~  A )
 
Theoreminfunabs 7766 An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A  /\  B 
 ~<_  A )  ->  ( A  u.  B )  ~~  A )
 
Theoreminfcda 7767 The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\  om  ~<_  A )  ->  ( A  +c  B ) 
 ~~  ( A  u.  B ) )
 
Theoreminfdif 7768 The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A  /\  B  ~<  A )  ->  ( A  \  B ) 
 ~~  A )
 
Theoreminfdif2 7769 Cardinality ordering for an infinite set difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\  om  ~<_  A )  ->  ( ( A  \  B )  ~<_  B  <->  A  ~<_  B )
 )
 
Theoreminfxpdom 7770 Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A  /\  B 
 ~<_  A )  ->  ( A  X.  B )  ~<_  A )
 
Theoreminfxpabs 7771 Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  ->  ( A  X.  B )  ~~  A )
 
Theoreminfunsdom1 7772 The union of two sets that are strictly dominated by the infinite set  X is also dominated by  X. This version of infunsdom 7773 assumes additionally that  A is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
 
Theoreminfunsdom 7773 The union of two sets that are strictly dominated by the infinite set  X is also strictly dominated by  X. (Contributed by Mario Carneiro, 3-May-2015.)
 |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A 
 ~<  X  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
 
Theoreminfxp 7774 Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/= 
 (/) ) )  ->  ( A  X.  B ) 
 ~~  ( A  u.  B ) )
 
Theorempwcdadom 7775 A property of dominance over a powerset, and a main lemma for gchac 8228. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ~P ( A  +c  A )  ~<_  ( A  +c  B ) 
 ->  ~P A  ~<_  B )
 
Theoreminfpss 7776* Every infinite set has an equinumerous proper subset. Exercise 7 of [TakeutiZaring] p. 91. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( om  ~<_  A  ->  E. x ( x  C.  A  /\  x  ~~  A ) )
 
Theoreminfmap2 7777* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8131 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( om  ~<_  A  /\  B 
 ~<_  A  /\  ( A 
 ^m  B )  e. 
 dom  card )  ->  ( A  ^m  B )  ~~  { x  |  ( x 
 C_  A  /\  x  ~~  B ) } )
 
2.6.10  The Ackermann bijection
 
Theoremackbij2lem1 7778 Lemma for ackbij2 7802. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  om  ->  ~P A  C_  ( ~P om  i^i  Fin )
 )
 
Theoremackbij1lem1 7779 Lemma for ackbij2 7802. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( -.  A  e.  B  ->  ( B  i^i  suc 
 A )  =  ( B  i^i  A ) )
 
Theoremackbij1lem2 7780 Lemma for ackbij2 7802. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  B  ->  ( B  i^i  suc  A )  =  ( { A }  u.  ( B  i^i  A ) ) )
 
Theoremackbij1lem3 7781 Lemma for ackbij2 7802. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  om  ->  A  e.  ( ~P
 om  i^i  Fin ) )
 
Theoremackbij1lem4 7782 Lemma for ackbij2 7802. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  ( A  e.  om  ->  { A }  e.  ( ~P om  i^i  Fin ) )
 
Theoremackbij1lem5 7783 Lemma for ackbij2 7802. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  ( A  e.  om  ->  ( card `  ~P suc  A )  =  ( ( card `  ~P A )  +o  ( card `  ~P A ) ) )
 
Theoremackbij1lem6 7784 Lemma for ackbij2 7802. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  e.  ( ~P om  i^i  Fin )
 )  ->  ( A  u.  B )  e.  ( ~P om  i^i  Fin )
 )
 
Theoremackbij1lem7 7785* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  ( ~P om  i^i  Fin )  ->  ( F `  A )  =  ( card ` 
 U_ y  e.  A  ( { y }  X.  ~P y ) ) )
 
Theoremackbij1lem8 7786* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  om  ->  ( F `  { A } )  =  ( card `  ~P A ) )
 
Theoremackbij1lem9 7787* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  e.  ( ~P om  i^i  Fin )  /\  ( A  i^i  B )  =  (/) )  ->  ( F `  ( A  u.  B ) )  =  ( ( F `
  A )  +o  ( F `  B ) ) )
 
Theoremackbij1lem10 7788* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  F : ( ~P om  i^i  Fin ) --> om
 
Theoremackbij1lem11 7789* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  C_  A )  ->  B  e.  ( ~P om  i^i  Fin )
 )
 
Theoremackbij1lem12 7790* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( B  e.  ( ~P om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  A )  C_  ( F `
  B ) )
 
Theoremackbij1lem13 7791* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( F `  (/) )  =  (/)
 
Theoremackbij1lem14 7792* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
 
Theoremackbij1lem15 7793* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  e.  ( ~P om  i^i  Fin ) )  /\  (
 c  e.  om  /\  c  e.  A  /\  -.  c  e.  B ) )  ->  -.  ( F `  ( A  i^i  suc  c ) )  =  ( F `  ( B  i^i  suc  c )
 ) )
 
Theoremackbij1lem16 7794* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( ( A  e.  ( ~P om  i^i  Fin )  /\  B  e.  ( ~P om  i^i  Fin )
 )  ->  ( ( F `  A )  =  ( F `  B )  ->  A  =  B ) )
 
Theoremackbij1lem17 7795* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  F : ( ~P om  i^i  Fin ) -1-1-> om
 
Theoremackbij1lem18 7796* Lemma for ackbij1 7797. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  ( ~P om  i^i  Fin )  ->  E. b  e.  ( ~P om  i^i  Fin )
 ( F `  b
 )  =  suc  ( F `  A ) )
 
Theoremackbij1 7797* The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  F : ( ~P om  i^i  Fin ) -1-1-onto-> om
 
Theoremackbij1b 7798* The Ackermann bijection, part 1b: the bijection from ackbij1 7797 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   =>    |-  ( A  e.  om  ->  ( F " ~P A )  =  ( card ` 
 ~P A ) )
 
Theoremackbij2lem2 7799* Lemma for ackbij2 7802. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   &    |-  G  =  ( x  e.  _V  |->  ( y  e. 
 ~P dom  x  |->  ( F `  ( x
 " y ) ) ) )   =>    |-  ( A  e.  om  ->  ( rec ( G ,  (/) ) `  A ) : ( R1 `  A ) -1-1-onto-> ( card `  ( R1 `  A ) ) )
 
Theoremackbij2lem3 7800* Lemma for ackbij2 7802. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   &    |-  G  =  ( x  e.  _V  |->  ( y  e. 
 ~P dom  x  |->  ( F `  ( x
 " y ) ) ) )   =>    |-  ( A  e.  om  ->  ( rec ( G ,  (/) ) `  A )  C_  ( rec ( G ,  (/) ) `  suc  A ) )
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