HomeHome Metamath Proof Explorer
Theorem List (p. 79 of 328)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21514)
  Hilbert Space Explorer  Hilbert Space Explorer
(21515-23037)
  Users' Mathboxes  Users' Mathboxes
(23038-32776)
 

Theorem List for Metamath Proof Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremkmlem10 7801* 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 7802* 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 7803* 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 7804* 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 7805* 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 7806* 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 7807* 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 7808* Equivalence of the Axiom of Choice and Maes' AC ackm 8108. The proof consists of lemmas kmlem1 7792 through kmlem16 7807 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 7771 with biid 227) 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 7809 Extend class definition to include cardinal number addition.
 class  +c
 
Definitiondf-cda 7810* Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 7812 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 7811 Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 +c  Fn  ( _V  X. 
 _V )
 
Theoremcdaval 7812 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 8189, carddom 8192, and cardsdom 8193. 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 7813 Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B )  ~<_  ( A  +c  B ) )
 
Theoremcdaun 7814 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 7815 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 7816 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 7817 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 7818 Adding and subtracting one gives back the original set. Similar to pncan 9073 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( B  e.  ( A  +c  1o )  ->  ( ( A  +c  1o )  \  { B } )  ~~  A )
 
Theorempm110.643 7819 1+1=2 for cardinal number addition, derived from pm54.43 7649 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 7581), but after applying definitions, our theorem is equivalent. The comment for cdaval 7812 explains why we use  ~~ instead of =. See pm110.643ALT 7820 for a shorter proof that doesn't use pm54.43 7649. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
 |-  ( 1o  +c  1o )  ~~  2o
 
Theorempm110.643ALT 7820 Alternate proof of pm110.643 7819. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1o  +c  1o )  ~~  2o
 
Theoremcda0en 7821 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 7822 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 7823 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 7824 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 7825 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 7826 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 7827 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 7828 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 7829 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 7830 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 7831 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 7832 The cardinal sum of two finite sets is finite. (Contributed by NM, 22-Oct-2004.)
 |-  ( ( A  ~<  om 
 /\  B  ~<  om )  ->  ( A  +c  B )  ~<  om )
 
Theoremcdainflem 7833 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 7834 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 7835 An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A )
 
Theorempwcda1 7836 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 7837 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 7838 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 7839 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 7840 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 7841 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 7842 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 7843 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 7844 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 7845 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 7846* Lemma for domtriom 8085. 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 7847 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 7848 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 7849 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 7850 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 7851 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 7852 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 7853 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 7854 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 7855 The union of two sets that are strictly dominated by the infinite set  X is also dominated by  X. This version of infunsdom 7856 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 7856 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 7857 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 7858 A property of dominance over a powerset, and a main lemma for gchac 8311. 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 7859* Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 7955. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( om  ~<_  A  ->  E. x ( x  C.  A  /\  x  ~~  A ) )
 
Theoreminfmap2 7860* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8214 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 7861 Lemma for ackbij2 7885. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  om  ->  ~P A  C_  ( ~P om  i^i  Fin )
 )
 
Theoremackbij1lem1 7862 Lemma for ackbij2 7885. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( -.  A  e.  B  ->  ( B  i^i  suc 
 A )  =  ( B  i^i  A ) )
 
Theoremackbij1lem2 7863 Lemma for ackbij2 7885. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  B  ->  ( B  i^i  suc  A )  =  ( { A }  u.  ( B  i^i  A ) ) )
 
Theoremackbij1lem3 7864 Lemma for ackbij2 7885. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  om  ->  A  e.  ( ~P
 om  i^i  Fin ) )
 
Theoremackbij1lem4 7865 Lemma for ackbij2 7885. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  ( A  e.  om  ->  { A }  e.  ( ~P om  i^i  Fin ) )
 
Theoremackbij1lem5 7866 Lemma for ackbij2 7885. (Contributed by Stefan O'Rear, 19-Nov-2014.)
 |-  ( A  e.  om  ->  ( card `  ~P suc  A )  =  ( ( card `  ~P A )  +o  ( card `  ~P A ) ) )
 
Theoremackbij1lem6 7867 Lemma for ackbij2 7885. (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 7868* Lemma for ackbij1 7880. (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 7869* Lemma for ackbij1 7880. (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 7870* Lemma for ackbij1 7880. (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 7871* Lemma for ackbij1 7880. (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 7872* Lemma for ackbij1 7880. (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 7873* Lemma for ackbij1 7880. (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 7874* Lemma for ackbij1 7880. (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 7875* Lemma for ackbij1 7880. (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 7876* Lemma for ackbij1 7880. (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 7877* Lemma for ackbij1 7880. (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 7878* Lemma for ackbij1 7880. (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 7879* Lemma for ackbij1 7880. (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 7880* 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 7881* The Ackermann bijection, part 1b: the bijection from ackbij1 7880 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 7882* Lemma for ackbij2 7885. (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 7883* Lemma for ackbij2 7885. (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 7884* Lemma for ackbij2 7885. (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 7885* 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 7886 The set of hereditarily finite sets is countable. See ackbij2 7885 for an explicit bijection that works without Infinity. See also r1omALT 8414. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( R1 `  om )  ~~  om
 
Theoremfictb 7887 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 7888* 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 7889* 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 7890 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 7891* 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 7892* 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 7893 Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)
 |-  ( cf `  (/) )  =  (/)
 
Theoremcardcf 7894 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 7895 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 7896 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 7897 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 7898 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 7899 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 7900* 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 ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32776
  Copyright terms: Public domain < Previous  Next >