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Theorem List for Metamath Proof Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcda1en 7801 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 7802 Adding and subtracting one gives back the original set. Similar to pncan 9057 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o )  \  { B } )  ~~  A )
 
Theorempm110.643 7803 1+1=2 for cardinal number addition, derived from pm54.43 7633 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 7565), but after applying definitions, our theorem is equivalent. The comment for cdaval 7796 explains why we use  ~~ instead of =. See pm110.643ALT 7804 for a shorter proof that doesn't use pm54.43 7633. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
 |-  ( 1o  +c  1o )  ~~  2o
 
Theorempm110.643ALT 7804 Alternate proof of pm110.643 7803. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1o  +c  1o )  ~~  2o
 
Theoremcda0en 7805 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 7806 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 7807 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 7808 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 7809 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 7810 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 7811 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 7812 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 7813 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 7814 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 7815 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 7816 The cardinal sum of two finite sets is finite. (Contributed by NM, 22-Oct-2004.)
 |-  ( ( A  ~<  om 
 /\  B  ~<  om )  ->  ( A  +c  B )  ~<  om )
 
Theoremcdainflem 7817 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 7818 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 7819 An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A )
 
Theorempwcda1 7820 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 7821 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 7822 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 7823 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 7824 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 7825 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 7826 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 7827 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 7828 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 7829 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 7830* Lemma for domtriom 8069. 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 7831 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 7832 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 7833 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 7834 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 7835 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 7836 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 7837 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 7838 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 7839 The union of two sets that are strictly dominated by the infinite set  X is also dominated by  X. This version of infunsdom 7840 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 7840 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 7841 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 7842 A property of dominance over a powerset, and a main lemma for gchac 8295. 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 7843* Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 7939. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( om  ~<_  A  ->  E. x ( x  C.  A  /\  x  ~~  A ) )
 
Theoreminfmap2 7844* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8198 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 7845 Lemma for ackbij2 7869. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  om  ->  ~P A  C_  ( ~P om  i^i  Fin )
 )
 
Theoremackbij1lem1 7846 Lemma for ackbij2 7869. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( -.  A  e.  B  ->  ( B  i^i  suc 
 A )  =  ( B  i^i  A ) )
 
Theoremackbij1lem2 7847 Lemma for ackbij2 7869. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  B  ->  ( B  i^i  suc  A )  =  ( { A }  u.  ( B  i^i  A ) ) )
 
Theoremackbij1lem3 7848 Lemma for ackbij2 7869. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  om  ->  A  e.  ( ~P
 om  i^i  Fin ) )
 
Theoremackbij1lem4 7849 Lemma for ackbij2 7869. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  ( A  e.  om  ->  { A }  e.  ( ~P om  i^i  Fin ) )
 
Theoremackbij1lem5 7850 Lemma for ackbij2 7869. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  ( A  e.  om  ->  ( card `  ~P suc  A )  =  ( ( card `  ~P A )  +o  ( card `  ~P A ) ) )
 
Theoremackbij1lem6 7851 Lemma for ackbij2 7869. (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 7852* Lemma for ackbij1 7864. (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 7853* Lemma for ackbij1 7864. (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 7854* Lemma for ackbij1 7864. (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 7855* Lemma for ackbij1 7864. (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 7856* Lemma for ackbij1 7864. (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 7857* Lemma for ackbij1 7864. (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 7858* Lemma for ackbij1 7864. (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 7859* Lemma for ackbij1 7864. (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 7860* Lemma for ackbij1 7864. (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 7861* Lemma for ackbij1 7864. (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 7862* Lemma for ackbij1 7864. (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 7863* Lemma for ackbij1 7864. (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 7864* 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 7865* The Ackermann bijection, part 1b: the bijection from ackbij1 7864 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 7866* Lemma for ackbij2 7869. (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 7867* Lemma for ackbij2 7869. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   &    |-  G  =  ( x  e.  _V  |->  ( y  e. 
 ~P dom  x  |->  ( F `  ( x
 " y ) ) ) )   =>    |-  ( A  e.  om  ->  ( rec ( G ,  (/) ) `  A )  C_  ( rec ( G ,  (/) ) `  suc  A ) )
 
Theoremackbij2lem4 7868* Lemma for ackbij2 7869. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   &    |-  G  =  ( x  e.  _V  |->  ( y  e. 
 ~P dom  x  |->  ( F `  ( x
 " y ) ) ) )   =>    |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  A ) )
 
Theoremackbij2 7869* The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card ` 
 U_ y  e.  x  ( { y }  X.  ~P y ) ) )   &    |-  G  =  ( x  e.  _V  |->  ( y  e. 
 ~P dom  x  |->  ( F `  ( x
 " y ) ) ) )   &    |-  H  =  U. ( rec ( G ,  (/) ) " om )   =>    |-  H : U. ( R1 " om ) -1-1-onto-> om
 
Theoremr1om 7870 The set of hereditarily finite sets is countable. See ackbij2 7869 for an explicit bijection that works without Infinity. See also r1omALT 8398. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( R1 `  om )  ~~  om
 
Theoremfictb 7871 A set is countable iff its collection of finite intersections is countable. (Contributed by Jeff Hankins, 24-Aug-2009.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  B  ->  ( A  ~<_  om  <->  ( fi `  A )  ~<_  om )
 )
 
2.6.11  Cofinality (without Axiom of Choice)
 
Theoremcflem 7872* A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set  A. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  e.  V  ->  E. x E. y
 ( x  =  (
 card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) )
 
Theoremcfval 7873* Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number  A is the cardinality (size) of the smallest unbounded subset  y of the ordinal number. Unbounded means that for every member of  A, there is a member of  y that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( A  e.  On  ->  ( cf `  A )  =  |^| { x  |  E. y ( x  =  ( card `  y
 )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
 
Theoremcff 7874 Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |- 
 cf : On --> On
 
Theoremcfub 7875* An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( cf `  A )  C_  |^| { x  |  E. y ( x  =  ( card `  y )  /\  ( y  C_  A  /\  A  C_  U. y ) ) }
 
Theoremcflm 7876* Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)
 |-  ( ( A  e.  B  /\  Lim  A )  ->  ( cf `  A )  =  |^| { x  |  E. y ( x  =  ( card `  y
 )  /\  ( y  C_  A  /\  A  =  U. y ) ) }
 )
 
Theoremcf0 7877 Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)
 |-  ( cf `  (/) )  =  (/)
 
Theoremcardcf 7878 Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( card `  ( cf `  A ) )  =  ( cf `  A )
 
Theoremcflecard 7879 Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( cf `  A )  C_  ( card `  A )
 
Theoremcfle 7880 Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( cf `  A )  C_  A
 
Theoremcfon 7881 The cofinality of any set is an ordinal (although it only makes sense when  A is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  ( cf `  A )  e.  On
 
Theoremcfeq0 7882 Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
 |-  ( A  e.  On  ->  ( ( cf `  A )  =  (/)  <->  A  =  (/) ) )
 
Theoremcfsuc 7883 Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
 |-  ( A  e.  On  ->  ( cf `  suc  A )  =  1o )
 
Theoremcff1 7884* There is always a map from  ( cf `  A
) to  A (this is a stronger condition than the definition, which only presupposes a map from some  y  ~~  ( cf `  A ). (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( A  e.  On  ->  E. f ( f : ( cf `  A ) -1-1-> A  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z  C_  ( f `
  w ) ) )
 
Theoremcfflb 7885* If there is a cofinal map from  B to  A, then  B is at least  ( cf `  A
). This theorem and cff1 7884 motivate the picture of  ( cf `  A
) as the greatest lower bound of the domain of cofinal maps into  A. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f
 ( f : B --> A  /\  A. z  e.  A  E. w  e.  B  z  C_  (
 f `  w )
 )  ->  ( cf `  A )  C_  B ) )
 
Theoremcfval2 7886* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( A  e.  On  ->  ( cf `  A )  =  |^|_ x  e. 
 { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  ( card `  x )
 )
 
Theoremcoflim 7887* A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  =  A 
 <-> 
 A. x  e.  A  E. y  e.  B  x  C_  y ) )
 
Theoremcflim3 7888* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  A  e.  _V   =>    |-  ( Lim  A  ->  ( cf `  A )  =  |^|_ x  e. 
 { x  e.  ~P A  |  U. x  =  A }  ( card `  x ) )
 
Theoremcflim2 7889 The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  A  e.  _V   =>    |-  ( Lim  A  <->  Lim  ( cf `  A ) )
 
Theoremcfom 7890 Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Proof shortened by Mario Carneiro, 11-Jun-2015.)
 |-  ( cf `  om )  =  om
 
Theoremcfss 7891* There is a cofinal subset of  A of cardinality  ( cf `  A
). (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( Lim  A  ->  E. x ( x 
 C_  A  /\  x  ~~  ( cf `  A )  /\  U. x  =  A ) )
 
Theoremcfslb 7892 Any cofinal subset of  A is at least as large as  ( cf `  A
). (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( Lim 
 A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A ) 
 ~<_  B )
 
Theoremcfslbn 7893 Any subset of  A smaller than its cofinality has union less than  A. (This is the contrapositive to cfslb 7892.) (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( Lim 
 A  /\  B  C_  A  /\  B  ~<  ( cf `  A ) )  ->  U. B  e.  A )
 
Theoremcfslb2n 7894* Any small collection of small subsets of  A cannot have union  A, where "small" means smaller than the cofinality. This is a stronger version of cfslb 7892. This is a common application of cofinality: under AC,  ( aleph `  1
) is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( Lim 
 A  /\  A. x  e.  B  ( x  C_  A  /\  x  ~<  ( cf `  A ) ) ) 
 ->  ( B  ~<  ( cf `  A )  ->  U. B  =/=  A ) )
 
Theoremcofsmo 7895* Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by by Mario Carneiro, 20-Mar-2013.)
 |-  C  =  { y  e.  B  |  A. w  e.  y  ( f `  w )  e.  (
 f `  y ) }   &    |-  K  =  |^| { x  e.  B  |  z  C_  ( f `  x ) }   &    |-  O  = OrdIso (  _E  ,  C )   =>    |-  ( ( Ord 
 A  /\  B  e.  On )  ->  ( E. f ( f : B --> A  /\  A. z  e.  A  E. w  e.  B  z  C_  (
 f `  w )
 )  ->  E. x  e.  suc  B E. g
 ( g : x --> A  /\  Smo  g  /\  A. z  e.  A  E. v  e.  x  z  C_  ( g `  v
 ) ) ) )
 
Theoremcfsmolem 7896* Lemma for cfsmo 7897. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  F  =  ( z  e.  _V  |->  ( ( g `  dom  z
 )  u.  U_ t  e.  dom  z  suc  (
 z `  t )
 ) )   &    |-  G  =  (recs ( F )  |`  ( cf `  A ) )   =>    |-  ( A  e.  On  ->  E. f ( f : ( cf `  A )
 --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z 
 C_  ( f `  w ) ) )
 
Theoremcfsmo 7897* The map in cff1 7884 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( A  e.  On  ->  E. f ( f : ( cf `  A )
 --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z 
 C_  ( f `  w ) ) )
 
Theoremcfcoflem 7898* Lemma for cfcof 7900, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f
 ( f : B --> A  /\  Smo  f  /\  A. x  e.  A  E. y  e.  B  x  C_  ( f `  y
 ) )  ->  ( cf `  A )  C_  ( cf `  B ) ) )
 
Theoremcoftr 7899* If there is a cofinal map from  B to  A and another from  C to  A, then there is also a cofinal map from  C to  B. Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 7900. (Contributed by Mario Carneiro, 16-Mar-2013.)
 |-  H  =  ( t  e.  C  |->  |^| { n  e.  B  |  ( g `
  t )  C_  ( f `  n ) } )   =>    |-  ( E. f ( f : B --> A  /\  Smo  f  /\  A. x  e.  A  E. y  e.  B  x  C_  (
 f `  y )
 )  ->  ( E. g ( g : C --> A  /\  A. z  e.  A  E. w  e.  C  z  C_  (
 g `  w )
 )  ->  E. h ( h : C --> B  /\  A. s  e.  B  E. w  e.  C  s  C_  ( h `  w ) ) ) )
 
Theoremcfcof 7900* If there is a cofinal map from  A to  B, then they have the same cofinality. This was used as Definition 11.1 of [TakeutiZaring] p. 100, who defines an equivalence relation cof  ( A ,  B ) and defines our  cf ( B ) as the minimum  B such that cof  ( A ,  B
). (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f
 ( f : B --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  B  z  C_  ( f `  w ) )  ->  ( cf `  A )  =  (
 cf `  B )
 ) )
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