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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hsmexlem3 8201* | Lemma for hsmex 8205. Clear hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g. using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) |
OrdIso OrdIso ^{*} har | ||
Theorem | hsmexlem4 8202* | Lemma for hsmex 8205. The core induction, establishing bounds on the order types of iterated unions of the initial set. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
har har OrdIso | ||
Theorem | hsmexlem5 8203* | Lemma for hsmex 8205. Combining the above constraints, along with itunitc 8194 and tcrank 7701, gives an effective constraint on the rank of . (Contributed by Stefan O'Rear, 14-Feb-2015.) |
har har OrdIso har | ||
Theorem | hsmexlem6 8204* | Lemmr for hsmex 8205. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
har har OrdIso | ||
Theorem | hsmex 8205* | The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 7453. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Theorem | hsmex2 8206* | The set of hereditary size-limited sets, assuming ax-reg 7453. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Theorem | hsmex3 8207* | The set of hereditary size-limited sets, assuming ax-reg 7453, using strict comparison (an easy corrolary by separation). (Contributed by Stefan O'Rear, 11-Feb-2015.) |
In this section we add the Axiom of Choice ax-ac 8232, as well as weaker forms such as the axiom of countable choice ax-cc 8208 and dependent choice ax-dc 8219. We introduce these weaker forms so that theorems that do not need the full power of the axiom of choice, but need more than simple ZF, can use these intermediate axioms instead. The combination of the Zermel-Fraenkel axioms and the axiom of choice is often abbreviated as ZFC. The axiom of choice is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. The axiom of choice does not satisfy those who wish to have a constructive proof (e.g., it will not satify intuitionist logic). Thus, we make it easy to identify which proofs depend on the axiom of choice or its weaker forms. | ||
Axiom | ax-cc 8208* | The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8251, but is weak enough that it can be proven using DC (see axcc 8231). It is, however, strictly stronger than ZF and cannot be proven in ZF. It states that any countable collection of non-empty sets must have a choice function. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Theorem | axcc2lem 8209* | Lemma for axcc2 8210. (Contributed by Mario Carneiro, 8-Feb-2013.) |
Theorem | axcc2 8210* | A possibly more useful version of ax-cc using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.) |
Theorem | axcc3 8211* | A possibly more useful version of ax-cc 8208 using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Mario Carneiro, 26-Dec-2014.) |
Theorem | axcc4 8212* | A version of axcc3 8211 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.) |
Theorem | acncc 8213 | An ax-cc 8208 equivalent: every set has choice sets of length . (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | axcc4dom 8214* | Relax the constraint on axcc4 8212 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.) |
Theorem | domtriomlem 8215* | Lemma for domtriom 8216. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Theorem | domtriom 8216 | Trichotomy of equinumerosity for , proven using CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 8087) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Theorem | fin41 8217 | Under countable choice, the IV-finite sets (Dedekind-finite) coincide with I-finite (finite in the usual sense) sets. (Contributed by Mario Carneiro, 16-May-2015.) |
Fin^{IV} | ||
Theorem | dominf 8218 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8208. See dominfac 8342 for a version proved from ax-ac 8232. The axiom of Regularity is used for this proof, via inf3lem6 7481, and its use is necessary: otherwise the set or (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Axiom | ax-dc 8219* | Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. Dependent choice is equivalent to the statement that every (nonempty) pruned tree has a branch. This axiom is redundant in ZFC; see axdc 8295. But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013.) |
Theorem | dcomex 8220 | The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.) |
Theorem | axdc2lem 8221* | Lemma for axdc2 8222. We construct a relation based on such that iff , and show that the "function" described by ax-dc 8219 can be restricted so that it is a real function (since the stated properties only show that it is the superset of a function). (Contributed by Mario Carneiro, 25-Jan-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Theorem | axdc2 8222* | An apparent strengthening of ax-dc 8219 (but derived from it) which shows that there is a denumerable sequence for any function that maps elements of a set to nonempty subsets of such that for all . The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.) |
Theorem | axdc3lem 8223* | The class of finite approximations to the DC sequence is a set. (We derive here the stronger statement that is a subset of a specific set, namely .) (Unnecessary distinct variable restrictions were removed by David Abernethy, 18-Mar-2014.) (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario Carneiro, 18-Mar-2014.) |
Theorem | axdc3lem2 8224* | Lemma for axdc3 8227. We have constructed a "candidate set" , which consists of all finite sequences that satisfy our property of interest, namely on its domain, but with the added constraint that . These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 8219 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (for some integer ). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 8219 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence , we can construct the sequence that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.) |
Theorem | axdc3lem3 8225* | Simple substitution lemma for axdc3 8227. (Contributed by Mario Carneiro, 27-Jan-2013.) |
Theorem | axdc3lem4 8226* | Lemma for axdc3 8227. We have constructed a "candidate set" , which consists of all finite sequences that satisfy our property of interest, namely on its domain, but with the added constraint that . These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 8219 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (for some integer ). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 8219 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that is nonempty, and that always maps to a nonempty subset of , so that we can apply axdc2 8222. See axdc3lem2 8224 for the rest of the proof. (Contributed by Mario Carneiro, 27-Jan-2013.) |
Theorem | axdc3 8227* | Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value . This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. (Contributed by Mario Carneiro, 27-Jan-2013.) |
Theorem | axdc4lem 8228* | Lemma for axdc4 8229. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Theorem | axdc4 8229* | A more general version of axdc3 8227 that allows the function to vary with . (Contributed by Mario Carneiro, 31-Jan-2013.) |
Theorem | axcclem 8230* | Lemma for axcc 8231. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Theorem | axcc 8231* | Although CC can be proven trivially using ac5 8251, we prove it here using DC. (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Feb-2013.) |
Axiom | ax-ac 8232* |
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 8235 for a more detailed explanation. Theorem ac2 8234 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 8239 is slightly shorter when the biconditional of ax-ac 8232 is expanded into implication and negation. In axac3 8237 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 8442 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 8266, ac5 8251, and ac7 8247. The Axiom of Regularity ax-reg 7453 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 7904. Equivalents to AC are the well-ordering theorem weth 8269 and Zorn's lemma zorn 8281. See ac4 8249 for comments about stronger versions of AC. In order to avoid uses of ax-reg 7453 for derivation of AC equivalents, we provide ax-ac2 8236 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8236 from ax-ac 8232 is shown by theorem axac2 8240, and the reverse derivation by axac 8241. Therefore, new proofs should normally use ax-ac2 8236 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.) |
Theorem | zfac 8233* | Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 8232. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.) |
Theorem | ac2 8234* | 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 8235 is easier to understand.) Note: aceq0 7892 shows the logical equivalence to ax-ac 8232. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.) |
Theorem | ac3 8235* |
Axiom of Choice using abbreviations. The logical equivalence to ax-ac 8232
can be established by chaining aceq0 7892 and aceq2 7893. A standard
textbook version of AC is derived from this one in dfac2a 7903, and this
version of AC is derived from the textbook version in dfac2 7904.
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 7466. The key theorem for this (used in the proof of dfac2 7904) is preleq 7465. 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 8236* | In order to avoid uses of ax-reg 7453 for derivation of AC equivalents, we provide ax-ac2 8236, 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 8239. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1551 available. The derivation of ax-ac2 8236 from ax-ac 8232 is shown by theorem axac2 8240, and the reverse derivation by axac 8241. Note that we use ax-reg 7453 to derive ax-ac 8232 from ax-ac2 8236, but not to derive ax-ac2 8236 from ax-ac 8232. (Contributed by NM, 19-Dec-2016.) |
Theorem | axac3 8237 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8236 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.) |
CHOICE | ||
Theorem | axac3OLD 8238 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac 8232 as an axiom. Obsolete as of 20-Dec-2016. (Contributed by Mario Carneiro, 6-May-2015.) (New usage is discouraged.) |
CHOICE | ||
Theorem | ackm 8239* |
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 7939. 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 8240* | Derive ax-ac2 8236 from ax-ac 8232. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | axac 8241* | Derive ax-ac 8232 from ax-ac2 8236. Note that ax-reg 7453 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.) |
Theorem | axaci 8242 | Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.) |
CHOICE | ||
Theorem | cardeqv 8243 | All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | numth3 8244 | All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Theorem | numth2 8245* | 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 8246* | 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 8247* | 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 8248* | 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 8249* |
Equivalent of Axiom of Choice. We do not insist that be a
function. However, theorem ac5 8251, 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 8265. (Contributed by NM, 21-Jul-1996.) |
Theorem | ac4c 8250* | Equivalent of Axiom of Choice (class version) (Contributed by NM, 10-Feb-1997.) |
Theorem | ac5 8251* | 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 8249. (Contributed by NM, 29-Aug-1999.) |
Theorem | ac5b 8252* | Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.) |
Theorem | ac6num 8253* | A version of ac6 8254 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Theorem | ac6 8254* | 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 8258, allows to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.) |
Theorem | ac6c4 8255* | Equivalent of Axiom of Choice. is a collection of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.) |
Theorem | ac6c5 8256* | 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 8257* | 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 8258* | Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 7710, we derive this strong version of ac6 8254 that doesn't require to be a set. (Contributed by NM, 4-Feb-2004.) |
Theorem | ac6n 8259* | Equivalent of Axiom of Choice. Contrapositive of ac6s 8258. (Contributed by NM, 10-Jun-2007.) |
Theorem | ac6s2 8260* | Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8261. (Contributed by NM, 29-Sep-2006.) |
Theorem | ac6s3 8261* | Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.) |
Theorem | ac6sg 8262* | ac6s 8258 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.) |
Theorem | ac6sf 8263* | Version of ac6 8254 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) |
Theorem | ac6s4 8264* | 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 8265* | 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 8266* | 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 8267* | 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 7708). (Contributed by NM, 29-Sep-2006.) |
Theorem | numthcor 8268* | Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.) |
Theorem | weth 8269* | 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 8270* | Lemma for zorn2 8280. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem2 8271* | Lemma for zorn2 8280. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem3 8272* | Lemma for zorn2 8280. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem4 8273* | Lemma for zorn2 8280. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem5 8274* | Lemma for zorn2 8280. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem6 8275* | Lemma for zorn2 8280. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2lem7 8276* | Lemma for zorn2 8280. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | zorn2g 8277* | Zorn's Lemma of [Monk1] p. 117. This version of zorn2 8280 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 8278* | 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 8281 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 8279* | Variant of Zorn's lemma zorng 8278 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 8280* | 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 8270 through zorn2lem7 8276; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8276. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | zorn 8281* | 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 8280 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.) |
[] | ||
Theorem | zornn0 8282* | Variant of Zorn's lemma zorn 8281 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 8283* | Lemma for ttukey 8292. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.) |
Theorem | ttukeylem2 8284* | Lemma for ttukey 8292. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) |
Theorem | ttukeylem3 8285* | Lemma for ttukey 8292. (Contributed by Mario Carneiro, 11-May-2015.) |
recs | ||
Theorem | ttukeylem4 8286* | Lemma for ttukey 8292. (Contributed by Mario Carneiro, 15-May-2015.) |
recs | ||
Theorem | ttukeylem5 8287* | Lemma for ttukey 8292. The function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.) |
recs | ||
Theorem | ttukeylem6 8288* | Lemma for ttukey 8292. (Contributed by Mario Carneiro, 15-May-2015.) |
recs | ||
Theorem | ttukeylem7 8289* | Lemma for ttukey 8292. (Contributed by Mario Carneiro, 15-May-2015.) |
recs | ||
Theorem | ttukey2g 8290* | The Teichmüller-Tukey Lemma ttukey 8292 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 8291* | The Teichmüller-Tukey Lemma ttukey 8292 stated with the "choice" as an antecedent (the hypothesis says that is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.) |
Theorem | ttukey 8292* | 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 8293* | Lemma for axdc 8295. (Contributed by Mario Carneiro, 25-Jan-2013.) |
Theorem | axdclem2 8294* | Lemma for axdc 8295. 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 8295* | This theorem derives ax-dc 8219 using ax-ac 8232 and ax-inf 7486. 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 8296 | An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8248. AC is not needed for finite sets - see fodomfi 7282. See also fodomnum 7831. (Contributed by NM, 23-Jul-2004.) |
Theorem | fodomg 8297 | An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.) |
Theorem | fodomb 8298* | 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 8299 | 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 8300* | Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.) |
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