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Theorem List for Metamath Proof Explorer - 8001-8100   *Has distinct variable group(s)
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Statement

Theoremkmlem16 8001* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.)

Theoremdfackm 8002* Equivalence of the Axiom of Choice and Maes' AC ackm 8301. The proof consists of lemmas kmlem1 7986 through kmlem16 8001 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 7965 with biid 228) 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

2.6.9  Cardinal number arithmetic

Syntaxccda 8003 Extend class definition to include cardinal number addition.

Definitiondf-cda 8004* Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 8006 for its value and a description. (Contributed by NM, 24-Sep-2004.)

Theoremcdafn 8005 Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremcdaval 8006 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 8382, carddom 8385, and cardsdom 8386. 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.)

Theoremcdaun 8008 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by NM, 5-Apr-2007.)

Theoremcdaen 8009 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.)

Theoremcdaenun 8010 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremcda1en 8011 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.)

Theoremcda1dif 8012 Adding and subtracting one gives back the original set. Similar to pncan 9267 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)

Theorempm110.643 8013 1+1=2 for cardinal number addition, derived from pm54.43 7843 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 7775), but after applying definitions, our theorem is equivalent. The comment for cdaval 8006 explains why we use instead of =. See pm110.643ALT 8014 for a shorter proof that doesn't use pm54.43 7843. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)

Theorempm110.643ALT 8014 Alternate proof of pm110.643 8013. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcda0en 8015 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.)

Theoremxp2cda 8016 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.)

Theoremcdacomen 8017 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.)

Theoremcdaassen 8018 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.)

Theoremxpcdaen 8019 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.)

Theoremmapcdaen 8020 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.)

Theorempwcdaen 8021 Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremcdadom1 8022 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.)

Theoremcdadom2 8023 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.)

Theoremcdadom3 8024 A set is dominated by its cardinal sum with another. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremcdaxpdom 8025 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.)

Theoremcdafi 8026 The cardinal sum of two finite sets is finite. (Contributed by NM, 22-Oct-2004.)

Theoremcdainflem 8027 Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.)

Theoremcdainf 8028 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.)

Theoreminfcda1 8029 An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)

Theorempwcda1 8030 The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.)

Theorempwcdaidm 8031 If the natural numbers inject into , then is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremcdalepw 8032 If is idempotent under cardinal sum and is dominated by the power set of , then so is the cardinal sum of and . (Contributed by Mario Carneiro, 15-May-2015.)

Theoremonacda 8033 The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)

Theoremcardacda 8034 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.)

Theoremcdanum 8035 The cardinal sum of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremunnum 8036 The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremnnacda 8037 The cardinal and ordinal sums of finite ordinals are equal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.)

Theoremficardun 8038 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.)

Theoremficardun2 8039 The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)

Theorempwsdompw 8040* Lemma for domtriom 8279. This is the equinumerosity version of the algebraic identity . (Contributed by Mario Carneiro, 7-Feb-2013.)

Theoremunctb 8041 The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)

Theoreminfcdaabs 8042 Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfunabs 8043 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.)

Theoreminfcda 8044 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.)

Theoreminfdif 8045 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.)

Theoreminfdif2 8046 Cardinality ordering for an infinite set difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfxpdom 8047 Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfxpabs 8048 Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfunsdom1 8049 The union of two sets that are strictly dominated by the infinite set is also dominated by . This version of infunsdom 8050 assumes additionally that is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theoreminfunsdom 8050 The union of two sets that are strictly dominated by the infinite set is also strictly dominated by . (Contributed by Mario Carneiro, 3-May-2015.)

Theoreminfxp 8051 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.)

Theorempwcdadom 8052 A property of dominance over a powerset, and a main lemma for gchac 8504. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)

Theoreminfpss 8053* Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 8149. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoreminfmap2 8054* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8407 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.)

2.6.10  The Ackermann bijection

Theoremackbij2lem1 8055 Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem1 8056 Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem2 8057 Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem3 8058 Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem4 8059 Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Theoremackbij1lem5 8060 Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Theoremackbij1lem6 8061 Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem7 8062* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremackbij1lem8 8063* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Theoremackbij1lem9 8064* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Theoremackbij1lem10 8065* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem11 8066* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem12 8067* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem13 8068* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem14 8069* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem15 8070* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem16 8071* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem17 8072* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem18 8073* Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1 8074* 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.)

Theoremackbij1b 8075* The Ackermann bijection, part 1b: the bijection from ackbij1 8074 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij2lem2 8076* Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij2lem3 8077* Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij2lem4 8078* Lemma for ackbij2 8079. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij2 8079* The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremr1om 8080 The set of hereditarily finite sets is countable. See ackbij2 8079 for an explicit bijection that works without Infinity. See also r1omALT 8607. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremfictb 8081 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.)

2.6.11  Cofinality (without Axiom of Choice)

Theoremcflem 8082* A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set . (Contributed by NM, 24-Apr-2004.)

Theoremcfval 8083* Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number is the cardinality (size) of the smallest unbounded subset of the ordinal number. Unbounded means that for every member of , there is a member of 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.)

Theoremcff 8084 Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theoremcfub 8085* An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremcflm 8086* Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)

Theoremcf0 8087 Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)

Theoremcardcf 8088 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.)

Theoremcflecard 8089 Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremcfle 8090 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.)

Theoremcfon 8091 The cofinality of any set is an ordinal (although it only makes sense when is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoremcfeq0 8092 Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)

Theoremcfsuc 8093 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.)

Theoremcff1 8094* There is always a map from to (this is a stronger condition than the definition, which only presupposes a map from some . (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcfflb 8095* If there is a cofinal map from to , then is at least . This theorem and cff1 8094 motivate the picture of as the greatest lower bound of the domain of cofinal maps into . (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcfval2 8096* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcoflim 8097* A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcflim3 8098* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcflim2 8099 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.)

Theoremcfom 8100 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.)

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