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Type | Label | Description |
---|---|---|
Statement | ||
Axiom | ax-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The unpublished version given here says that given any set , there exists a that is a collection of unordered pairs, one pair for each non-empty member of .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.) |
Theorem | zfac 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.) |
Theorem | ac2 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.) |
Theorem | ac3 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.) |
Axiom | ax-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.) |
Theorem | axac3 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 | ||
Theorem | axac3OLD 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 | ||
Theorem | ackm 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.) |
Theorem | axac2 8109* | Derive ax-ac2 8105 from ax-ac 8101. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | axac 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.) |
Theorem | axaci 8111 | Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.) |
CHOICE | ||
Theorem | cardeqv 8112 | All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | numth3 8113 | All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Theorem | numth2 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.) |
Theorem | numth 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.) |
Theorem | ac7 8116* | An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.) |
Theorem | ac7g 8117* | An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.) |
Theorem | ac4 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.) |
Theorem | ac4c 8119* | Equivalent of Axiom of Choice (class version) (Contributed by NM, 10-Feb-1997.) |
Theorem | ac5 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.) |
Theorem | ac5b 8121* | Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.) |
Theorem | ac6num 8122* | A version of ac6 8123 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Theorem | ac6 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.) |
Theorem | ac6c4 8124* | Equivalent of Axiom of Choice. is a collection of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.) |
Theorem | ac6c5 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.) |
Theorem | ac9 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.) |
Theorem | ac6s 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.) |
Theorem | ac6n 8128* | Equivalent of Axiom of Choice. Contrapositive of ac6s 8127. (Contributed by NM, 10-Jun-2007.) |
Theorem | ac6s2 8129* | Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8130. (Contributed by NM, 29-Sep-2006.) |
Theorem | ac6s3 8130* | Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.) |
Theorem | ac6sg 8131* | ac6s 8127 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.) |
Theorem | ac6sf 8132* | Version of ac6 8123 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) |
Theorem | ac6s4 8133* | Generalization of the Axiom of Choice to proper classes. is a collection of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.) |
Theorem | ac6s5 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.) |
Theorem | ac8 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.) |
Theorem | ac9s 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.) |
Theorem | numthcor 8137* | Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.) |
Theorem | weth 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.) |
Theorem | zorn2lem1 8139* | Lemma for zorn2 8149. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem2 8140* | Lemma for zorn2 8149. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem3 8141* | Lemma for zorn2 8149. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem4 8142* | Lemma for zorn2 8149. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem5 8143* | Lemma for zorn2 8149. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem6 8144* | Lemma for zorn2 8149. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem7 8145* | Lemma for zorn2 8149. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2g 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.) |
Theorem | zorng 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.) |
[] | ||
Theorem | zornn0g 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.) |
[] | ||
Theorem | zorn2 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.) |
Theorem | zorn 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.) |
[] | ||
Theorem | zornn0 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.) |
[] | ||
Theorem | ttukeylem1 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.) |
Theorem | ttukeylem2 8153* | Lemma for ttukey 8161. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) |
Theorem | ttukeylem3 8154* | Lemma for ttukey 8161. (Contributed by Mario Carneiro, 11-May-2015.) |
recs | ||
Theorem | ttukeylem4 8155* | Lemma for ttukey 8161. (Contributed by Mario Carneiro, 15-May-2015.) |
recs | ||
Theorem | ttukeylem5 8156* | Lemma for ttukey 8161. The function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.) |
recs | ||
Theorem | ttukeylem6 8157* | Lemma for ttukey 8161. (Contributed by Mario Carneiro, 15-May-2015.) |
recs | ||
Theorem | ttukeylem7 8158* | Lemma for ttukey 8161. (Contributed by Mario Carneiro, 15-May-2015.) |
recs | ||
Theorem | ttukey2g 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.) |
Theorem | ttukeyg 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.) |
Theorem | ttukey 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.) |
Theorem | axdclem 8162* | Lemma for axdc 8164. (Contributed by Mario Carneiro, 25-Jan-2013.) |
Theorem | axdclem2 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.) |
Theorem | axdc 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.) |
Theorem | fodom 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.) |
Theorem | fodomg 8166 | An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.) |
Theorem | fodomb 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.) |
Theorem | wdomac 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.) |
^{*} | ||
Theorem | brdom3 8169* | Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.) |
Theorem | brdom5 8170* | An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.) |
Theorem | brdom4 8171* | An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
Theorem | brdom7disj 8172* | An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
Theorem | brdom6disj 8173* | An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.) |
Theorem | fin71ac 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.) |
Fin^{VII} | ||
Theorem | imadomg 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.) |
Theorem | fnrndomg 8176 | The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.) |
Theorem | iunfo 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.) |
Theorem | iundom2g 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 | ||
Theorem | iundomg 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 | ||
Theorem | iundom 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.) |
Theorem | unidom 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.) |
Theorem | uniimadom 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.) |
Theorem | uniimadomf 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.) |
Theorem | cardval 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.) |
Theorem | cardid 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.) |
Theorem | cardidg 8186 | Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8185. (Contributed by David Moews, 1-May-2017.) |
Theorem | cardidd 8187 | Any set is equinumerous to its cardinal number. Deduction form of cardid 8185. (Contributed by David Moews, 1-May-2017.) |
Theorem | cardf 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.) |
Theorem | carden 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.) |
Theorem | cardeq0 8190 | Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.) |
Theorem | unsnen 8191 | Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.) |
Theorem | carddom 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.) |
Theorem | cardsdom 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.) |
Theorem | domtri 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.) |
Theorem | entric 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.) |
Theorem | entri2 8196 | Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.) |
Theorem | entri3 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.) |
Theorem | sdomsdomcard 8198 | A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.) |
Theorem | canth3 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.) |
Theorem | infxpidm 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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