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

Theoremaceq0 8001* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 8341. (Contributed by NM, 5-Apr-2004.)

Theoremaceq2 8002* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)

Theoremaceq3lem 8003* Lemma for dfac3 8004. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremdfac3 8004* Equivalence of two versions of the Axiom of Choice. The left-hand side is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Stefan O'Rear, 22-Feb-2015.)
CHOICE

Theoremdfac4 8005* Equivalence of two versions of the Axiom of Choice. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
CHOICE

Theoremdfac5lem1 8006* Lemma for dfac5 8011. (Contributed by NM, 12-Apr-2004.)

Theoremdfac5lem2 8007* Lemma for dfac5 8011. (Contributed by NM, 12-Apr-2004.)

Theoremdfac5lem3 8008* Lemma for dfac5 8011. (Contributed by NM, 12-Apr-2004.)

Theoremdfac5lem4 8009* Lemma for dfac5 8011. (Contributed by NM, 11-Apr-2004.)

Theoremdfac5lem5 8010* Lemma for dfac5 8011. (Contributed by NM, 12-Apr-2004.)

Theoremdfac5 8011* Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
CHOICE

Theoremdfac2a 8012* Our Axiom of Choice (in the form of ac3 8344) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 8013 for the converse (which does use the Axiom of Regularity). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
CHOICE

Theoremdfac2 8013* Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 8344). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 7567 and preleq 7574 that are referenced in the proof each make use of Regularity for their derivations. (The reverse implication can be derived without using Regularity; see dfac2a 8012.) TODO: Fix label in comment, and put label changes into list at top of set.mm. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
CHOICE

Theoremdfac7 8014* Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 8343). The proof does not depend AC on but does depend on the Axiom of Regularity. (Contributed by Mario Carneiro, 17-May-2015.)
CHOICE

Theoremdfac0 8015* Equivalence of two versions of the Axiom of Choice. The proof uses the Axiom of Regularity. The right-hand side is our original ax-ac 8341. (Contributed by Mario Carneiro, 17-May-2015.)
CHOICE

Theoremdfac1 8016* Equivalence of two versions of the Axiom of Choice ax-ac 8341. The proof uses the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by Mario Carneiro, 17-May-2015.)
CHOICE

Theoremdfac8 8017* A proof of the equivalency of the Well Ordering Theorem weth 8377 and the Axiom of Choice ac7 8355. (Contributed by Mario Carneiro, 5-Jan-2013.)
CHOICE

Theoremdfac9 8018* Equivalence of the axiom of choice with a statement related to ac9 8365; definition AC3 of [Schechter] p. 139. (Contributed by Stefan O'Rear, 22-Feb-2015.)
CHOICE

Theoremdfac10 8019 Axiom of Choice equivalent: the cardinality function measures every set. (Contributed by Mario Carneiro, 6-May-2015.)
CHOICE

Theoremdfac10c 8020* Axiom of Choice equivalent: every set is equinumerous to an ordinal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
CHOICE

Theoremdfac10b 8021 Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 7999). (Contributed by Stefan O'Rear, 17-Jan-2015.)
CHOICE

Theoremacacni 8022 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
CHOICE AC

Theoremdfacacn 8023 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
CHOICE AC

Theoremdfac13 8024 The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
CHOICE AC

Theoremdfac12lem1 8025* Lemma for dfac12 8031. (Contributed by Mario Carneiro, 29-May-2015.)
har       recs OrdIso               OrdIso

Theoremdfac12lem2 8026* Lemma for dfac12 8031. (Contributed by Mario Carneiro, 29-May-2015.)
har       recs OrdIso               OrdIso

Theoremdfac12lem3 8027* Lemma for dfac12 8031. (Contributed by Mario Carneiro, 29-May-2015.)
har       recs OrdIso

Theoremdfac12r 8028 The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 8031 does not assume the Axiom of Regularity. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremdfac12k 8029* Equivalence of dfac12 8031 and dfac12a 8030, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)

Theoremdfac12a 8030 The axiom of choice holds iff every ordinal has a well-orderable powerset. (Contributed by Mario Carneiro, 29-May-2015.)
CHOICE

Theoremdfac12 8031 The axiom of choice holds iff every aleph has a well-orderable powerset. (Contributed by Mario Carneiro, 21-May-2015.)
CHOICE

Theoremkmlem1 8032* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2. (Contributed by NM, 5-Apr-2004.)

Theoremkmlem2 8033* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Theoremkmlem3 8034* 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.)

Theoremkmlem4 8035* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)

Theoremkmlem5 8036* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Theoremkmlem6 8037* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)

Theoremkmlem7 8038* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)

Theoremkmlem8 8039* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 4-Apr-2004.)

Theoremkmlem9 8040* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Theoremkmlem10 8041* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Theoremkmlem11 8042* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)

Theoremkmlem12 8043* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.)

Theoremkmlem13 8044* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 5-Apr-2004.)

Theoremkmlem14 8045* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)

Theoremkmlem15 8046* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)

Theoremkmlem16 8047* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.)

Theoremdfackm 8048* Equivalence of the Axiom of Choice and Maes' AC ackm 8347. The proof consists of lemmas kmlem1 8032 through kmlem16 8047 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 8011 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

2.6.9  Cardinal number arithmetic

Syntaxccda 8049 Extend class definition to include cardinal number addition.

Definitiondf-cda 8050* Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 8052 for its value and a description. (Contributed by NM, 24-Sep-2004.)

Theoremcdafn 8051 Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremcdaval 8052 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 8428, carddom 8431, and cardsdom 8432. 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 8054 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by NM, 5-Apr-2007.)

Theoremcdaen 8055 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 8056 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremcda1en 8057 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 8058 Adding and subtracting one gives back the original set. Similar to pncan 9313 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)

Theorempm110.643 8059 1+1=2 for cardinal number addition, derived from pm54.43 7889 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 7821), but after applying definitions, our theorem is equivalent. The comment for cdaval 8052 explains why we use instead of =. See pm110.643ALT 8060 for a shorter proof that doesn't use pm54.43 7889. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)

Theorempm110.643ALT 8060 Alternate proof of pm110.643 8059. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcda0en 8061 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 8062 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 8063 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 8064 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 8065 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 8066 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 8067 Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremcdadom1 8068 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 8069 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 8070 A set is dominated by its cardinal sum with another. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremcdaxpdom 8071 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 8072 The cardinal sum of two finite sets is finite. (Contributed by NM, 22-Oct-2004.)

Theoremcdainflem 8073 Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.)

Theoremcdainf 8074 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 8075 An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)

Theorempwcda1 8076 The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.)

Theorempwcdaidm 8077 If the natural numbers inject into , then is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremcdalepw 8078 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 8079 The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)

Theoremcardacda 8080 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 8081 The cardinal sum of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremunnum 8082 The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremnnacda 8083 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 8084 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 8085 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 8086* Lemma for domtriom 8325. This is the equinumerosity version of the algebraic identity . (Contributed by Mario Carneiro, 7-Feb-2013.)

Theoremunctb 8087 The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)

Theoreminfcdaabs 8088 Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfunabs 8089 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 8090 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 8091 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 8092 Cardinality ordering for an infinite set difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfxpdom 8093 Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfxpabs 8094 Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfunsdom1 8095 The union of two sets that are strictly dominated by the infinite set is also dominated by . This version of infunsdom 8096 assumes additionally that is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theoreminfunsdom 8096 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 8097 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 8098 A property of dominance over a powerset, and a main lemma for gchac 8550. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)

Theoreminfpss 8099* Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 8195. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoreminfmap2 8100* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8453 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.)

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