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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
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Statement

3.2  ZFC Set Theory - add the Axiom of Choice

3.2.1  Introduce the Axiom of Choice

Axiomax-ac 8101* Axiom of Choice. The Axiom of Choice (AC) is usually considered an extension of ZF set theory rather than a proper part of it. It is sometimes considered philosophically controversial because it asserts the existence of a set without telling us what the set is. ZF set theory that includes AC is called ZFC.

The unpublished version given here says that given any set , there exists a that is a collection of unordered pairs, one pair for each non-empty member of . One entry in the pair is the member of , and the other entry is some arbitrary member of that member of . See the rewritten version ac3 8104 for a more detailed explanation. Theorem ac2 8103 shows an equivalent written compactly with restricted quantifiers.

This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 8108 is slightly shorter when the biconditional of ax-ac 8101 is expanded into implication and negation. In axac3 8106 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 8311 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8135, ac5 8120, and ac7 8116. The Axiom of Regularity ax-reg 7322 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 7773. Equivalents to AC are the well-ordering theorem weth 8138 and Zorn's lemma zorn 8150. See ac4 8118 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 7322 for derivation of AC equivalents, we provide ax-ac2 8105 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8105 from ax-ac 8101 is shown by theorem axac2 8109, and the reverse derivation by axac 8110. Therefore, new proofs should normally use ax-ac2 8105 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

Theoremzfac 8102* Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 8101. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)

Theoremac2 8103* Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single explicit conjunction. (If you want to figure it out, the rewritten equivalent ac3 8104 is easier to understand.) Note: aceq0 7761 shows the logical equivalence to ax-ac 8101. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

Theoremac3 8104* Axiom of Choice using abbreviations. The logical equivalence to ax-ac 8101 can be established by chaining aceq0 7761 and aceq2 7762. A standard textbook version of AC is derived from this one in dfac2a 7772, and this version of AC is derived from the textbook version in dfac2 7773.

The following sketch will help you understand this version of the axiom. Given any set , the axiom says that there exists a that is a collection of unordered pairs, one pair for each non-empty member of . One entry in the pair is the member of , and the other entry is some arbitrary member of that member of . Using the Axiom of Regularity, we can show that is really a set of ordered pairs, very similar to the ordered pair construction opthreg 7335. The key theorem for this (used in the proof of dfac2 7773) is preleq 7334. With this modified definition of ordered pair, it can be seen that is actually a choice function on the members of .

For example, suppose . Let us try . For the member (of ) , the only assignment to and that satisfies the axiom is and , so there is exactly one as required. We verify the other two members of similarly. Thus, satisfies the axiom. Using our modified ordered pair definition, we can say that corresponds to the choice function . Of course other choices for will also satisfy the axiom, for example . What AC tells us is that there exists at least one such , but it doesn't tell us which one.

(New usage is discouraged.) (Contributed by NM, 19-Jul-1996.)

Axiomax-ac2 8105* In order to avoid uses of ax-reg 7322 for derivation of AC equivalents, we provide ax-ac2 8105, which is equivalent to the standard AC of textbooks. This appears to be the shortest known equivalent to the standard AC when expressed in terms of set theory primitives. It was found by Kurt Maes as theorem ackm 8108. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1536 available. The derivation of ax-ac2 8105 from ax-ac 8101 is shown by theorem axac2 8109, and the reverse derivation by axac 8110. Note that we use ax-reg 7322 to derive ax-ac 8101 from ax-ac2 8105, but not to derive ax-ac2 8105 from ax-ac 8101. (Contributed by NM, 19-Dec-2016.)

Theoremaxac3 8106 This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8105 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
CHOICE

Theoremaxac3OLD 8107 This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac 8101 as an axiom. Obsolete as of 20-Dec-2016. (Contributed by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
CHOICE

Theoremackm 8108* A remarkable equivalent to the Axiom of Choice that has only 5 quantifiers (when expanded to , primitives in prenex form), discovered and proved by Kurt Maes. This establishes a new record, reducing from 6 to 5 the largest number of quantified variables needed by any ZFC axiom. The ZF-equivalence to AC is shown by theorem dfackm 7808. Maes found this version of AC in April, 2004 (replacing a longer version, also with 5 quantifiers, that he found in November, 2003). See Kurt Maes, "A 5-quantifier (,=)-expression ZF-equivalent to the Axiom of Choice" (http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3162v1.pdf).

The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)

Theoremaxac2 8109* Derive ax-ac2 8105 from ax-ac 8101. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremaxac 8110* Derive ax-ac 8101 from ax-ac2 8105. Note that ax-reg 7322 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)

Theoremaxaci 8111 Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
CHOICE

Theoremcardeqv 8112 All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremnumth3 8113 All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremnumth2 8114* Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.)

Theoremnumth 8115* Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)

Theoremac7 8116* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.)

Theoremac7g 8117* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)

Theoremac4 8118* Equivalent of Axiom of Choice. We do not insist that be a function. However, theorem ac5 8120, derived from this one, shows that this form of the axiom does imply that at least one such set whose existence we assert is in fact a function. Axiom of Choice of [TakeutiZaring] p. 83.

Takeuti and Zaring call this "weak choice" in contrast to "strong choice" , which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable and thus cannot be expressed in our first-order formalization. However, it has been shown that ZF plus strong choice is a conservative extension of ZF plus weak choice. See Ulrich Felgner, "Comparison of the axioms of local and universal choice," Fundamenta Mathematica, 71, 43-62 (1971).

Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 8134. (Contributed by NM, 21-Jul-1996.)

Theoremac4c 8119* Equivalent of Axiom of Choice (class version) (Contributed by NM, 10-Feb-1997.)

Theoremac5 8120* An Axiom of Choice equivalent: there exists a function (called a choice function) with domain that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that be a function is not necessary; see ac4 8118. (Contributed by NM, 29-Aug-1999.)

Theoremac5b 8121* Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)

Theoremac6num 8122* A version of ac6 8123 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)

Theoremac6 8123* Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set , where depends on (the natural number) and (to specify a member of ). A stronger version of this theorem, ac6s 8127, allows to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)

Theoremac6c4 8124* Equivalent of Axiom of Choice. is a collection of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.)

Theoremac6c5 8125* Equivalent of Axiom of Choice. is a collection of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)

Theoremac9 8126* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.)

Theoremac6s 8127* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 7579, we derive this strong version of ac6 8123 that doesn't require to be a set. (Contributed by NM, 4-Feb-2004.)

Theoremac6n 8128* Equivalent of Axiom of Choice. Contrapositive of ac6s 8127. (Contributed by NM, 10-Jun-2007.)

Theoremac6s2 8129* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8130. (Contributed by NM, 29-Sep-2006.)

Theoremac6s3 8130* Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)

Theoremac6sg 8131* ac6s 8127 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)

Theoremac6sf 8132* Version of ac6 8123 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)

Theoremac6s4 8133* Generalization of the Axiom of Choice to proper classes. is a collection of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.)

Theoremac6s5 8134* Generalization of the Axiom of Choice to proper classes. is a collection of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.)

Theoremac8 8135* An Axiom of Choice equivalent. Given a family of mutually disjoint nonempty sets, there exists a set containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.)

Theoremac9s 8136* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. This is a stronger version of the axiom in Enderton, with no existence requirement for the family of classes (achieved via the Collection Principle cp 7577). (Contributed by NM, 29-Sep-2006.)

3.2.2  AC equivalents: well-ordering, Zorn's lemma

Theoremnumthcor 8137* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)

Theoremweth 8138* Well-ordering theorem: any set can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario Carneiro, 5-Jan-2013.)

Theoremzorn2lem1 8139* Lemma for zorn2 8149. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem2 8140* Lemma for zorn2 8149. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem3 8141* Lemma for zorn2 8149. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem4 8142* Lemma for zorn2 8149. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem5 8143* Lemma for zorn2 8149. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem6 8144* Lemma for zorn2 8149. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem7 8145* Lemma for zorn2 8149. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2g 8146* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 8149 avoids the Axiom of Choice by assuming that is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremzorng 8147* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 8150 avoids the Axiom of Choice by assuming that is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
[]

Theoremzornn0g 8148* Variant of Zorn's lemma zorng 8147 in which , the union of the empty chain, is not required to be an element of . (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)
[]

Theoremzorn2 8149* Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set (with an ordering relation ) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 8139 through zorn2lem7 8145; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8145. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremzorn 8150* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 8149 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
[]

Theoremzornn0 8151* Variant of Zorn's lemma zorn 8150 in which , the union of the empty chain, is not required to be an element of . (Contributed by Jeff Madsen, 5-Jan-2011.)
[]

Theoremttukeylem1 8152* Lemma for ttukey 8161. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremttukeylem2 8153* Lemma for ttukey 8161. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremttukeylem3 8154* Lemma for ttukey 8161. (Contributed by Mario Carneiro, 11-May-2015.)
recs

Theoremttukeylem4 8155* Lemma for ttukey 8161. (Contributed by Mario Carneiro, 15-May-2015.)
recs

Theoremttukeylem5 8156* Lemma for ttukey 8161. The function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
recs

Theoremttukeylem6 8157* Lemma for ttukey 8161. (Contributed by Mario Carneiro, 15-May-2015.)
recs

Theoremttukeylem7 8158* Lemma for ttukey 8161. (Contributed by Mario Carneiro, 15-May-2015.)
recs

Theoremttukey2g 8159* The Teichmüller-Tukey Lemma ttukey 8161 with a slightly stronger conclusion: we can set up the maximal element of so that it also contains some given as a subset. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremttukeyg 8160* The Teichmüller-Tukey Lemma ttukey 8161 stated with the "choice" as an antecedent (the hypothesis says that is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.)

Theoremttukey 8161* The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If is a nonempty collection of finite character, then has a maximal element with respect to inclusion. Here "finite character" means that iff every finite subset of is in . (Contributed by Mario Carneiro, 15-May-2015.)

Theoremaxdclem 8162* Lemma for axdc 8164. (Contributed by Mario Carneiro, 25-Jan-2013.)

Theoremaxdclem2 8163* Lemma for axdc 8164. Using the full Axiom of Choice, we can construct a choice function on . From this, we can build a sequence starting at any value by repeatedly applying to the set (where is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)

Theoremaxdc 8164* This theorem derives ax-dc 8088 using ax-ac 8101 and ax-inf 7355. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.)

Theoremfodom 8165 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8117. AC is not needed for finite sets - see fodomfi 7151. See also fodomnum 7700. (Contributed by NM, 23-Jul-2004.)

Theoremfodomg 8166 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)

Theoremfodomb 8167* Equivalence of an onto mapping and dominance for a non-empty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)

Theoremwdomac 8168 When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theorembrdom3 8169* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)

Theorembrdom5 8170* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)

Theorembrdom4 8171* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theorembrdom7disj 8172* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theorembrdom6disj 8173* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)

Theoremfin71ac 8174 Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.)
FinVII

Theoremimadomg 8175 An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)

Theoremfnrndomg 8176 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)

Theoremiunfo 8177* Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)

Theoremiundom2g 8178* An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. depends on and should be thought of as . (Contributed by Mario Carneiro, 1-Sep-2015.)
AC

Theoremiundomg 8179* An upper bound for the cardinality of an indexed union, with explicit choice principles. depends on and should be thought of as . (Contributed by Mario Carneiro, 1-Sep-2015.)
AC               AC

Theoremiundom 8180* An upper bound for the cardinality of an indexed union. depends on and should be thought of as . (Contributed by NM, 26-Mar-2006.)

Theoremunidom 8181* An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98. (Contributed by NM, 25-Mar-2006.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremuniimadom 8182* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.)

Theoremuniimadomf 8183* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8182 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)

3.2.3  Cardinal number theorems using Axiom of Choice

Theoremcardval 8184* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 7640 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremcardid 8185 Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremcardidg 8186 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8185. (Contributed by David Moews, 1-May-2017.)

Theoremcardidd 8187 Any set is equinumerous to its cardinal number. Deduction form of cardid 8185. (Contributed by David Moews, 1-May-2017.)

Theoremcardf 8188 The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremcarden 8189 Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size." This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank (see karden 7581). (Contributed by NM, 22-Oct-2003.)

Theoremcardeq0 8190 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)

Theoremunsnen 8191 Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)

Theoremcarddom 8192 Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremcardsdom 8193 Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremdomtri 8194 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theorementric 8195 Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242. (Contributed by NM, 4-Jan-2004.)

Theorementri2 8196 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)

Theorementri3 8197 Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275. (Contributed by NM, 4-Jan-2004.)

Theoremsdomsdomcard 8198 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)

Theoremcanth3 8199 Cantor's theorem in terms of cardinals. This theorem tells us that no matter how largei a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.)

Theoreminfxpidm 8200 The cross product of an infinite set with itself is idempotent. This theorem (which is an AC equivalent) provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This proof follows as a corollary of infxpen 7658. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)

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