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Type | Label | Description |
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Statement | ||
Theorem | hsmexlem6 10301* | Lemma for hsmex 10302. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
β’ π β V & β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) & β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) & β’ π = {π β βͺ (π 1 β On) β£ βπ β (TCβ{π})π βΌ π} & β’ π = OrdIso( E , (rank β ((πβπ)βπ))) β β’ π β V | ||
Theorem | hsmex 10302* | 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 9462. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
β’ (π β π β {π β βͺ (π 1 β On) β£ βπ₯ β (TCβ{π })π₯ βΌ π} β V) | ||
Theorem | hsmex2 10303* | The set of hereditary size-limited sets, assuming ax-reg 9462. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
β’ (π β π β {π β£ βπ₯ β (TCβ{π })π₯ βΌ π} β V) | ||
Theorem | hsmex3 10304* | The set of hereditary size-limited sets, assuming ax-reg 9462, using strict comparison (an easy corollary by separation). (Contributed by Stefan O'Rear, 11-Feb-2015.) |
β’ (π β π β {π β£ βπ₯ β (TCβ{π })π₯ βΊ π} β V) | ||
In this section we add the Axiom of Choice ax-ac 10329, as well as weaker forms such as the axiom of countable choice ax-cc 10305 and dependent choice ax-dc 10316. 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 Zermelo-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 satisfy intuitionistic logic). Thus, we make it easy to identify which proofs depend on the axiom of choice or its weaker forms. | ||
Axiom | ax-cc 10305* | The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10347, but is weak enough that it can be proven using DC (see axcc 10328). It is, however, strictly stronger than ZF and cannot be proven in ZF. It states that any countable collection of nonempty sets must have a choice function. (Contributed by Mario Carneiro, 9-Feb-2013.) |
β’ (π₯ β Ο β βπβπ§ β π₯ (π§ β β β (πβπ§) β π§)) | ||
Theorem | axcc2lem 10306* | Lemma for axcc2 10307. (Contributed by Mario Carneiro, 8-Feb-2013.) |
β’ πΎ = (π β Ο β¦ if((πΉβπ) = β , {β }, (πΉβπ))) & β’ π΄ = (π β Ο β¦ ({π} Γ (πΎβπ))) & β’ πΊ = (π β Ο β¦ (2nd β(πβ(π΄βπ)))) β β’ βπ(π Fn Ο β§ βπ β Ο ((πΉβπ) β β β (πβπ) β (πΉβπ))) | ||
Theorem | axcc2 10307* | 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.) |
β’ βπ(π Fn Ο β§ βπ β Ο ((πΉβπ) β β β (πβπ) β (πΉβπ))) | ||
Theorem | axcc3 10308* | A possibly more useful version of ax-cc 10305 using sequences πΉ(π) 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.) |
β’ πΉ β V & β’ π β Ο β β’ βπ(π Fn π β§ βπ β π (πΉ β β β (πβπ) β πΉ)) | ||
Theorem | axcc4 10309* | A version of axcc3 10308 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.) |
β’ π΄ β V & β’ π β Ο & β’ (π₯ = (πβπ) β (π β π)) β β’ (βπ β π βπ₯ β π΄ π β βπ(π:πβΆπ΄ β§ βπ β π π)) | ||
Theorem | acncc 10310 | An ax-cc 10305 equivalent: every set has choice sets of length Ο. (Contributed by Mario Carneiro, 31-Aug-2015.) |
β’ AC Ο = V | ||
Theorem | axcc4dom 10311* | Relax the constraint on axcc4 10309 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.) |
β’ π΄ β V & β’ (π₯ = (πβπ) β (π β π)) β β’ ((π βΌ Ο β§ βπ β π βπ₯ β π΄ π) β βπ(π:πβΆπ΄ β§ βπ β π π)) | ||
Theorem | domtriomlem 10312* | Lemma for domtriom 10313. (Contributed by Mario Carneiro, 9-Feb-2013.) |
β’ π΄ β V & β’ π΅ = {π¦ β£ (π¦ β π΄ β§ π¦ β π« π)} & β’ πΆ = (π β Ο β¦ ((πβπ) β βͺ π β π (πβπ))) β β’ (Β¬ π΄ β Fin β Ο βΌ π΄) | ||
Theorem | domtriom 10313 | Trichotomy of equinumerosity for Ο, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 10184) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.) |
β’ π΄ β V β β’ (Ο βΌ π΄ β Β¬ π΄ βΊ Ο) | ||
Theorem | fin41 10314 | 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.) |
β’ FinIV = Fin | ||
Theorem | dominf 10315 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 10305. See dominfac 10443 for a version proved from ax-ac 10329. The axiom of Regularity is used for this proof, via inf3lem6 9503, and its use is necessary: otherwise the set π΄ = {π΄} or π΄ = {β , π΄} (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.) |
β’ π΄ β V β β’ ((π΄ β β β§ π΄ β βͺ π΄) β Ο βΌ π΄) | ||
Axiom | ax-dc 10316* | 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 10391. 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.) |
β’ ((βπ¦βπ§ π¦π₯π§ β§ ran π₯ β dom π₯) β βπβπ β Ο (πβπ)π₯(πβsuc π)) | ||
Theorem | dcomex 10317 | 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.) |
β’ Ο β V | ||
Theorem | axdc2lem 10318* | Lemma for axdc2 10319. We construct a relation π based on πΉ such that π₯π π¦ iff π¦ β (πΉβπ₯), and show that the "function" described by ax-dc 10316 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.) |
β’ π΄ β V & β’ π = {β¨π₯, π¦β© β£ (π₯ β π΄ β§ π¦ β (πΉβπ₯))} & β’ πΊ = (π₯ β Ο β¦ (ββπ₯)) β β’ ((π΄ β β β§ πΉ:π΄βΆ(π« π΄ β {β })) β βπ(π:ΟβΆπ΄ β§ βπ β Ο (πβsuc π) β (πΉβ(πβπ)))) | ||
Theorem | axdc2 10319* | An apparent strengthening of ax-dc 10316 (but derived from it) which shows that there is a denumerable sequence π for any function that maps elements of a set π΄ to nonempty subsets of π΄ such that π(π₯ + 1) β πΉ(π(π₯)) for all π₯ β Ο. 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.) |
β’ π΄ β V β β’ ((π΄ β β β§ πΉ:π΄βΆ(π« π΄ β {β })) β βπ(π:ΟβΆπ΄ β§ βπ β Ο (πβsuc π) β (πΉβ(πβπ)))) | ||
Theorem | axdc3lem 10320* | The class π of finite approximations to the DC sequence is a set. (We derive here the stronger statement that π is a subset of a specific set, namely π« (Ο Γ π΄).) (Contributed by Mario Carneiro, 27-Jan-2013.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 18-Mar-2014.) |
β’ π΄ β V & β’ π = {π β£ βπ β Ο (π :suc πβΆπ΄ β§ (π ββ ) = πΆ β§ βπ β π (π βsuc π) β (πΉβ(π βπ)))} β β’ π β V | ||
Theorem | axdc3lem2 10321* | Lemma for axdc3 10324. We have constructed a "candidate set" π, which consists of all finite sequences π that satisfy our property of interest, namely π (π₯ + 1) β πΉ(π (π₯)) on its domain, but with the added constraint that π (0) = πΆ. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 10316 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (ββπ):πβΆπ΄ (for some integer π). 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 10316 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 β, we can construct the sequence π that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.) |
β’ π΄ β V & β’ π = {π β£ βπ β Ο (π :suc πβΆπ΄ β§ (π ββ ) = πΆ β§ βπ β π (π βsuc π) β (πΉβ(π βπ)))} & β’ πΊ = (π₯ β π β¦ {π¦ β π β£ (dom π¦ = suc dom π₯ β§ (π¦ βΎ dom π₯) = π₯)}) β β’ (ββ(β:ΟβΆπ β§ βπ β Ο (ββsuc π) β (πΊβ(ββπ))) β βπ(π:ΟβΆπ΄ β§ (πββ ) = πΆ β§ βπ β Ο (πβsuc π) β (πΉβ(πβπ)))) | ||
Theorem | axdc3lem3 10322* | Simple substitution lemma for axdc3 10324. (Contributed by Mario Carneiro, 27-Jan-2013.) |
β’ π΄ β V & β’ π = {π β£ βπ β Ο (π :suc πβΆπ΄ β§ (π ββ ) = πΆ β§ βπ β π (π βsuc π) β (πΉβ(π βπ)))} & β’ π΅ β V β β’ (π΅ β π β βπ β Ο (π΅:suc πβΆπ΄ β§ (π΅ββ ) = πΆ β§ βπ β π (π΅βsuc π) β (πΉβ(π΅βπ)))) | ||
Theorem | axdc3lem4 10323* | Lemma for axdc3 10324. We have constructed a "candidate set" π, which consists of all finite sequences π that satisfy our property of interest, namely π (π₯ + 1) β πΉ(π (π₯)) on its domain, but with the added constraint that π (0) = πΆ. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 10316 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (ββπ):πβΆπ΄ (for some integer π). 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 10316 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 π is nonempty, and that πΊ always maps to a nonempty subset of π, so that we can apply axdc2 10319. See axdc3lem2 10321 for the rest of the proof. (Contributed by Mario Carneiro, 27-Jan-2013.) |
β’ π΄ β V & β’ π = {π β£ βπ β Ο (π :suc πβΆπ΄ β§ (π ββ ) = πΆ β§ βπ β π (π βsuc π) β (πΉβ(π βπ)))} & β’ πΊ = (π₯ β π β¦ {π¦ β π β£ (dom π¦ = suc dom π₯ β§ (π¦ βΎ dom π₯) = π₯)}) β β’ ((πΆ β π΄ β§ πΉ:π΄βΆ(π« π΄ β {β })) β βπ(π:ΟβΆπ΄ β§ (πββ ) = πΆ β§ βπ β Ο (πβsuc π) β (πΉβ(πβπ)))) | ||
Theorem | axdc3 10324* | Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value πΆ. 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.) |
β’ π΄ β V β β’ ((πΆ β π΄ β§ πΉ:π΄βΆ(π« π΄ β {β })) β βπ(π:ΟβΆπ΄ β§ (πββ ) = πΆ β§ βπ β Ο (πβsuc π) β (πΉβ(πβπ)))) | ||
Theorem | axdc4lem 10325* | Lemma for axdc4 10326. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.) |
β’ π΄ β V & β’ πΊ = (π β Ο, π₯ β π΄ β¦ ({suc π} Γ (ππΉπ₯))) β β’ ((πΆ β π΄ β§ πΉ:(Ο Γ π΄)βΆ(π« π΄ β {β })) β βπ(π:ΟβΆπ΄ β§ (πββ ) = πΆ β§ βπ β Ο (πβsuc π) β (ππΉ(πβπ)))) | ||
Theorem | axdc4 10326* | A more general version of axdc3 10324 that allows the function πΉ to vary with π. (Contributed by Mario Carneiro, 31-Jan-2013.) |
β’ π΄ β V β β’ ((πΆ β π΄ β§ πΉ:(Ο Γ π΄)βΆ(π« π΄ β {β })) β βπ(π:ΟβΆπ΄ β§ (πββ ) = πΆ β§ βπ β Ο (πβsuc π) β (ππΉ(πβπ)))) | ||
Theorem | axcclem 10327* | Lemma for axcc 10328. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.) |
β’ π΄ = (π₯ β {β }) & β’ πΉ = (π β Ο, π¦ β βͺ π΄ β¦ (πβπ)) & β’ πΊ = (π€ β π΄ β¦ (ββsuc (β‘πβπ€))) β β’ (π₯ β Ο β βπβπ§ β π₯ (π§ β β β (πβπ§) β π§)) | ||
Theorem | axcc 10328* | Although CC can be proven trivially using ac5 10347, we prove it here using DC. (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Feb-2013.) |
β’ (π₯ β Ο β βπβπ§ β π₯ (π§ β β β (πβπ§) β π§)) | ||
Axiom | ax-ac 10329* |
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 π₯, there exists a π¦ that is a collection of unordered pairs, one pair for each nonempty member of π₯. One entry in the pair is the member of π₯, and the other entry is some arbitrary member of that member of π₯. See the rewritten version ac3 10332 for a more detailed explanation. Theorem ac2 10331 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 10335 is slightly shorter when the biconditional of ax-ac 10329 is expanded into implication and negation. In axac3 10334 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10551 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 10362, ac5 10347, and ac7 10343. The Axiom of Regularity ax-reg 9462 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as Theorem dfac2b 10000. Equivalents to AC are the well-ordering theorem weth 10365 and Zorn's lemma zorn 10377. See ac4 10345 for comments about stronger versions of AC. In order to avoid uses of ax-reg 9462 for derivation of AC equivalents, we provide ax-ac2 10333 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 10333 from ax-ac 10329 is shown by Theorem axac2 10336, and the reverse derivation by axac 10337. Therefore, new proofs should normally use ax-ac2 10333 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.) |
β’ βπ¦βπ§βπ€((π§ β π€ β§ π€ β π₯) β βπ£βπ’(βπ‘((π’ β π€ β§ π€ β π‘) β§ (π’ β π‘ β§ π‘ β π¦)) β π’ = π£)) | ||
Theorem | zfac 10330* | Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 10329. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.) |
β’ βπ₯βπ¦βπ§((π¦ β π§ β§ π§ β π€) β βπ€βπ¦(βπ€((π¦ β π§ β§ π§ β π€) β§ (π¦ β π€ β§ π€ β π₯)) β π¦ = π€)) | ||
Theorem | ac2 10331* | 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 10332 is easier to understand.) Note: aceq0 9988 shows the logical equivalence to ax-ac 10329. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.) |
β’ βπ¦βπ§ β π₯ βπ€ β π§ β!π£ β π§ βπ’ β π¦ (π§ β π’ β§ π£ β π’) | ||
Theorem | ac3 10332* |
Axiom of Choice using abbreviations. The logical equivalence to ax-ac 10329
can be established by chaining aceq0 9988 and aceq2 9989. A standard
textbook version of AC is derived from this one in dfac2a 9999, and this
version of AC is derived from the textbook version in dfac2b 10000, showing
their logical equivalence (see dfac2 10001).
The following sketch will help you understand this version of the axiom. Given any set π₯, the axiom says that there exists a π¦ that is a collection of unordered pairs, one pair for each nonempty member of π₯. One entry in the pair is the member of π₯, and the other entry is some arbitrary member of that member of π₯. Using the Axiom of Regularity, we can show that π¦ is really a set of ordered pairs, very similar to the ordered pair construction opthreg 9488. The key theorem for this (used in the proof of dfac2b 10000) is preleq 9486. With this modified definition of ordered pair, it can be seen that π¦ is actually a choice function on the members of π₯. For example, suppose π₯ = {{1, 2}, {1, 3}, {2, 3, 4}}. Let us try π¦ = {{{1, 2}, 1}, {{1, 3}, 1}, {{2, 3, 4}, 2}}. For the member (of π₯) π§ = {1, 2}, the only assignment to π€ and π£ that satisfies the axiom is π€ = 1 and π£ = {{1, 2}, 1}, so there is exactly one π€ as required. We verify the other two members of π₯ similarly. Thus, π¦ satisfies the axiom. Using our modified ordered pair definition, we can say that π¦ corresponds to the choice function {β¨{1, 2}, 1β©, β¨{1, 3}, 1β©, β¨{2, 3, 4}, 2β©}. Of course other choices for π¦ will also satisfy the axiom, for example π¦ = {{{1, 2}, 2}, {{1, 3}, 1}, {{2, 3, 4}, 4}}. What AC tells us is that there exists at least one such π¦, but it doesn't tell us which one. (New usage is discouraged.) (Contributed by NM, 19-Jul-1996.) |
β’ βπ¦βπ§ β π₯ (π§ β β β β!π€ β π§ βπ£ β π¦ (π§ β π£ β§ π€ β π£)) | ||
Axiom | ax-ac2 10333* | In order to avoid uses of ax-reg 9462 for derivation of AC equivalents, we provide ax-ac2 10333, 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 10335. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1798 available. The derivation of ax-ac2 10333 from ax-ac 10329 is shown by Theorem axac2 10336, and the reverse derivation by axac 10337. Note that we use ax-reg 9462 to derive ax-ac 10329 from ax-ac2 10333, but not to derive ax-ac2 10333 from ax-ac 10329. (Contributed by NM, 19-Dec-2016.) |
β’ βπ¦βπ§βπ£βπ’((π¦ β π₯ β§ (π§ β π¦ β ((π£ β π₯ β§ Β¬ π¦ = π£) β§ π§ β π£))) β¨ (Β¬ π¦ β π₯ β§ (π§ β π₯ β ((π£ β π§ β§ π£ β π¦) β§ ((π’ β π§ β§ π’ β π¦) β π’ = π£))))) | ||
Theorem | axac3 10334 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10333 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.) |
β’ CHOICE | ||
Theorem | ackm 10335* |
A remarkable equivalent to the Axiom of Choice that has only five
quantifiers (when expanded to use only the primitive predicates =
and β and in prenex normal 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 10036. Maes found this
version of AC in April 2004 (replacing a longer version, also with five
quantifiers, that he found in November 2003). See Kurt Maes, "A
5-quantifier (β , =)-expression
ZF-equivalent to the Axiom of
Choice", https://doi.org/10.48550/arXiv.0705.3162 10036.
The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html 10036 http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html 10036. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.) |
β’ βπ₯βπ¦βπ§βπ£βπ’((π¦ β π₯ β§ (π§ β π¦ β ((π£ β π₯ β§ Β¬ π¦ = π£) β§ π§ β π£))) β¨ (Β¬ π¦ β π₯ β§ (π§ β π₯ β ((π£ β π§ β§ π£ β π¦) β§ ((π’ β π§ β§ π’ β π¦) β π’ = π£))))) | ||
Theorem | axac2 10336* | Derive ax-ac2 10333 from ax-ac 10329. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ βπ¦βπ§βπ£βπ’((π¦ β π₯ β§ (π§ β π¦ β ((π£ β π₯ β§ Β¬ π¦ = π£) β§ π§ β π£))) β¨ (Β¬ π¦ β π₯ β§ (π§ β π₯ β ((π£ β π§ β§ π£ β π¦) β§ ((π’ β π§ β§ π’ β π¦) β π’ = π£))))) | ||
Theorem | axac 10337* | Derive ax-ac 10329 from ax-ac2 10333. Note that ax-reg 9462 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.) |
β’ βπ¦βπ§βπ€((π§ β π€ β§ π€ β π₯) β βπ£βπ’(βπ‘((π’ β π€ β§ π€ β π‘) β§ (π’ β π‘ β§ π‘ β π¦)) β π’ = π£)) | ||
Theorem | axaci 10338 | Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ (CHOICE β βπ₯π) β β’ π | ||
Theorem | cardeqv 10339 | All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.) |
β’ dom card = V | ||
Theorem | numth3 10340 | All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
β’ (π΄ β π β π΄ β dom card) | ||
Theorem | numth2 10341* | Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.) |
β’ π΄ β V β β’ βπ₯ β On π₯ β π΄ | ||
Theorem | numth 10342* | 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.) |
β’ π΄ β V β β’ βπ₯ β On βπ π:π₯β1-1-ontoβπ΄ | ||
Theorem | ac7 10343* | An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.) |
β’ βπ(π β π₯ β§ π Fn dom π₯) | ||
Theorem | ac7g 10344* | An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.) |
β’ (π β π΄ β βπ(π β π β§ π Fn dom π )) | ||
Theorem | ac4 10345* |
Equivalent of Axiom of Choice. We do not insist that π be a
function. However, Theorem ac5 10347, derived from this one, shows that
this form of the axiom does imply that at least one such set π 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" βπΉβπ§(π§ β β β (πΉβπ§) β π§), which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable πΉ 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 10361. (Contributed by NM, 21-Jul-1996.) |
β’ βπβπ§ β π₯ (π§ β β β (πβπ§) β π§) | ||
Theorem | ac4c 10346* | Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.) |
β’ π΄ β V β β’ βπβπ₯ β π΄ (π₯ β β β (πβπ₯) β π₯) | ||
Theorem | ac5 10347* | An Axiom of Choice equivalent: there exists a function π (called a choice function) with domain π΄ 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 π be a function is not necessary; see ac4 10345. (Contributed by NM, 29-Aug-1999.) |
β’ π΄ β V β β’ βπ(π Fn π΄ β§ βπ₯ β π΄ (π₯ β β β (πβπ₯) β π₯)) | ||
Theorem | ac5b 10348* | Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.) |
β’ π΄ β V β β’ (βπ₯ β π΄ π₯ β β β βπ(π:π΄βΆβͺ π΄ β§ βπ₯ β π΄ (πβπ₯) β π₯)) | ||
Theorem | ac6num 10349* | A version of ac6 10350 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.) |
β’ (π¦ = (πβπ₯) β (π β π)) β β’ ((π΄ β π β§ βͺ π₯ β π΄ {π¦ β π΅ β£ π} β dom card β§ βπ₯ β π΄ βπ¦ β π΅ π) β βπ(π:π΄βΆπ΅ β§ βπ₯ β π΄ π)) | ||
Theorem | ac6 10350* | 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 π΅, where π depends on π₯ (the natural number) and π¦ (to specify a member of π΅). A stronger version of this theorem, ac6s 10354, allows π΅ to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.) |
β’ π΄ β V & β’ π΅ β V & β’ (π¦ = (πβπ₯) β (π β π)) β β’ (βπ₯ β π΄ βπ¦ β π΅ π β βπ(π:π΄βΆπ΅ β§ βπ₯ β π΄ π)) | ||
Theorem | ac6c4 10351* | Equivalent of Axiom of Choice. π΅ is a collection π΅(π₯) of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.) |
β’ π΄ β V & β’ π΅ β V β β’ (βπ₯ β π΄ π΅ β β β βπ(π Fn π΄ β§ βπ₯ β π΄ (πβπ₯) β π΅)) | ||
Theorem | ac6c5 10352* | Equivalent of Axiom of Choice. π΅ is a collection π΅(π₯) of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.) |
β’ π΄ β V & β’ π΅ β V β β’ (βπ₯ β π΄ π΅ β β β βπβπ₯ β π΄ (πβπ₯) β π΅) | ||
Theorem | ac9 10353* | 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.) |
β’ π΄ β V & β’ π΅ β V β β’ (βπ₯ β π΄ π΅ β β β Xπ₯ β π΄ π΅ β β ) | ||
Theorem | ac6s 10354* | Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9763, we derive this strong version of ac6 10350 that doesn't require π΅ to be a set. (Contributed by NM, 4-Feb-2004.) |
β’ π΄ β V & β’ (π¦ = (πβπ₯) β (π β π)) β β’ (βπ₯ β π΄ βπ¦ β π΅ π β βπ(π:π΄βΆπ΅ β§ βπ₯ β π΄ π)) | ||
Theorem | ac6n 10355* | Equivalent of Axiom of Choice. Contrapositive of ac6s 10354. (Contributed by NM, 10-Jun-2007.) |
β’ π΄ β V & β’ (π¦ = (πβπ₯) β (π β π)) β β’ (βπ(π:π΄βΆπ΅ β βπ₯ β π΄ π) β βπ₯ β π΄ βπ¦ β π΅ π) | ||
Theorem | ac6s2 10356* | Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 10357. (Contributed by NM, 29-Sep-2006.) |
β’ π΄ β V & β’ (π¦ = (πβπ₯) β (π β π)) β β’ (βπ₯ β π΄ βπ¦π β βπ(π Fn π΄ β§ βπ₯ β π΄ π)) | ||
Theorem | ac6s3 10357* | Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.) |
β’ π΄ β V & β’ (π¦ = (πβπ₯) β (π β π)) β β’ (βπ₯ β π΄ βπ¦π β βπβπ₯ β π΄ π) | ||
Theorem | ac6sg 10358* | ac6s 10354 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.) |
β’ (π¦ = (πβπ₯) β (π β π)) β β’ (π΄ β π β (βπ₯ β π΄ βπ¦ β π΅ π β βπ(π:π΄βΆπ΅ β§ βπ₯ β π΄ π))) | ||
Theorem | ac6sf 10359* | Version of ac6 10350 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) |
β’ β²π¦π & β’ π΄ β V & β’ (π¦ = (πβπ₯) β (π β π)) β β’ (βπ₯ β π΄ βπ¦ β π΅ π β βπ(π:π΄βΆπ΅ β§ βπ₯ β π΄ π)) | ||
Theorem | ac6s4 10360* | Generalization of the Axiom of Choice to proper classes. π΅ is a collection π΅(π₯) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.) |
β’ π΄ β V β β’ (βπ₯ β π΄ π΅ β β β βπ(π Fn π΄ β§ βπ₯ β π΄ (πβπ₯) β π΅)) | ||
Theorem | ac6s5 10361* | Generalization of the Axiom of Choice to proper classes. π΅ is a collection π΅(π₯) of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.) |
β’ π΄ β V β β’ (βπ₯ β π΄ π΅ β β β βπβπ₯ β π΄ (πβπ₯) β π΅) | ||
Theorem | ac8 10362* | An Axiom of Choice equivalent. Given a family π₯ of mutually disjoint nonempty sets, there exists a set π¦ containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.) |
β’ ((βπ§ β π₯ π§ β β β§ βπ§ β π₯ βπ€ β π₯ (π§ β π€ β (π§ β© π€) = β )) β βπ¦βπ§ β π₯ β!π£ π£ β (π§ β© π¦)) | ||
Theorem | ac9s 10363* | 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 π΅(π₯) (achieved via the Collection Principle cp 9761). (Contributed by NM, 29-Sep-2006.) |
β’ π΄ β V β β’ (βπ₯ β π΄ π΅ β β β Xπ₯ β π΄ π΅ β β ) | ||
Theorem | numthcor 10364* | Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.) |
β’ (π΄ β π β βπ₯ β On π΄ βΊ π₯) | ||
Theorem | weth 10365* | Well-ordering theorem: any set π΄ 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.) |
β’ (π΄ β π β βπ₯ π₯ We π΄) | ||
Theorem | zorn2lem1 10366* | Lemma for zorn2 10376. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ πΉ = recs((π β V β¦ (β©π£ β πΆ βπ’ β πΆ Β¬ π’π€π£))) & β’ πΆ = {π§ β π΄ β£ βπ β ran π ππ π§} & β’ π· = {π§ β π΄ β£ βπ β (πΉ β π₯)ππ π§} β β’ ((π₯ β On β§ (π€ We π΄ β§ π· β β )) β (πΉβπ₯) β π·) | ||
Theorem | zorn2lem2 10367* | Lemma for zorn2 10376. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ πΉ = recs((π β V β¦ (β©π£ β πΆ βπ’ β πΆ Β¬ π’π€π£))) & β’ πΆ = {π§ β π΄ β£ βπ β ran π ππ π§} & β’ π· = {π§ β π΄ β£ βπ β (πΉ β π₯)ππ π§} β β’ ((π₯ β On β§ (π€ We π΄ β§ π· β β )) β (π¦ β π₯ β (πΉβπ¦)π (πΉβπ₯))) | ||
Theorem | zorn2lem3 10368* | Lemma for zorn2 10376. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ πΉ = recs((π β V β¦ (β©π£ β πΆ βπ’ β πΆ Β¬ π’π€π£))) & β’ πΆ = {π§ β π΄ β£ βπ β ran π ππ π§} & β’ π· = {π§ β π΄ β£ βπ β (πΉ β π₯)ππ π§} β β’ ((π Po π΄ β§ (π₯ β On β§ (π€ We π΄ β§ π· β β ))) β (π¦ β π₯ β Β¬ (πΉβπ₯) = (πΉβπ¦))) | ||
Theorem | zorn2lem4 10369* | Lemma for zorn2 10376. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ πΉ = recs((π β V β¦ (β©π£ β πΆ βπ’ β πΆ Β¬ π’π€π£))) & β’ πΆ = {π§ β π΄ β£ βπ β ran π ππ π§} & β’ π· = {π§ β π΄ β£ βπ β (πΉ β π₯)ππ π§} β β’ ((π Po π΄ β§ π€ We π΄) β βπ₯ β On π· = β ) | ||
Theorem | zorn2lem5 10370* | Lemma for zorn2 10376. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ πΉ = recs((π β V β¦ (β©π£ β πΆ βπ’ β πΆ Β¬ π’π€π£))) & β’ πΆ = {π§ β π΄ β£ βπ β ran π ππ π§} & β’ π· = {π§ β π΄ β£ βπ β (πΉ β π₯)ππ π§} & β’ π» = {π§ β π΄ β£ βπ β (πΉ β π¦)ππ π§} β β’ (((π€ We π΄ β§ π₯ β On) β§ βπ¦ β π₯ π» β β ) β (πΉ β π₯) β π΄) | ||
Theorem | zorn2lem6 10371* | Lemma for zorn2 10376. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ πΉ = recs((π β V β¦ (β©π£ β πΆ βπ’ β πΆ Β¬ π’π€π£))) & β’ πΆ = {π§ β π΄ β£ βπ β ran π ππ π§} & β’ π· = {π§ β π΄ β£ βπ β (πΉ β π₯)ππ π§} & β’ π» = {π§ β π΄ β£ βπ β (πΉ β π¦)ππ π§} β β’ (π Po π΄ β (((π€ We π΄ β§ π₯ β On) β§ βπ¦ β π₯ π» β β ) β π Or (πΉ β π₯))) | ||
Theorem | zorn2lem7 10372* | Lemma for zorn2 10376. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ πΉ = recs((π β V β¦ (β©π£ β πΆ βπ’ β πΆ Β¬ π’π€π£))) & β’ πΆ = {π§ β π΄ β£ βπ β ran π ππ π§} & β’ π· = {π§ β π΄ β£ βπ β (πΉ β π₯)ππ π§} & β’ π» = {π§ β π΄ β£ βπ β (πΉ β π¦)ππ π§} β β’ ((π΄ β dom card β§ π Po π΄ β§ βπ ((π β π΄ β§ π Or π ) β βπ β π΄ βπ β π (ππ π β¨ π = π))) β βπ β π΄ βπ β π΄ Β¬ ππ π) | ||
Theorem | zorn2g 10373* | Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10376 avoids the Axiom of Choice by assuming that π΄ is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ ((π΄ β dom card β§ π Po π΄ β§ βπ€((π€ β π΄ β§ π Or π€) β βπ₯ β π΄ βπ§ β π€ (π§π π₯ β¨ π§ = π₯))) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯π π¦) | ||
Theorem | zorng 10374* | 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 10377 avoids the Axiom of Choice by assuming that π΄ is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ ((π΄ β dom card β§ βπ§((π§ β π΄ β§ [β] Or π§) β βͺ π§ β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | zornn0g 10375* | Variant of Zorn's lemma zorng 10374 in which β , the union of the empty chain, is not required to be an element of π΄. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ ((π΄ β dom card β§ π΄ β β β§ βπ§((π§ β π΄ β§ π§ β β β§ [β] Or π§) β βͺ π§ β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | zorn2 10376* | Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set π΄ (with an ordering relation π ) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 10366 through zorn2lem7 10372; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 10372. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ π΄ β V β β’ ((π Po π΄ β§ βπ€((π€ β π΄ β§ π Or π€) β βπ₯ β π΄ βπ§ β π€ (π§π π₯ β¨ π§ = π₯))) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯π π¦) | ||
Theorem | zorn 10377* | 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 10376 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.) |
β’ π΄ β V β β’ (βπ§((π§ β π΄ β§ [β] Or π§) β βͺ π§ β π΄) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | zornn0 10378* | Variant of Zorn's lemma zorn 10377 in which β , the union of the empty chain, is not required to be an element of π΄. (Contributed by Jeff Madsen, 5-Jan-2011.) |
β’ π΄ β V β β’ ((π΄ β β β§ βπ§((π§ β π΄ β§ π§ β β β§ [β] Or π§) β βͺ π§ β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | ttukeylem1 10379* | Lemma for ttukey 10388. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β β’ (π β (πΆ β π΄ β (π« πΆ β© Fin) β π΄)) | ||
Theorem | ttukeylem2 10380* | Lemma for ttukey 10388. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β β’ ((π β§ (πΆ β π΄ β§ π· β πΆ)) β π· β π΄) | ||
Theorem | ttukeylem3 10381* | Lemma for ttukey 10388. (Contributed by Mario Carneiro, 11-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ ((π β§ πΆ β On) β (πΊβπΆ) = if(πΆ = βͺ πΆ, if(πΆ = β , π΅, βͺ (πΊ β πΆ)), ((πΊββͺ πΆ) βͺ if(((πΊββͺ πΆ) βͺ {(πΉββͺ πΆ)}) β π΄, {(πΉββͺ πΆ)}, β )))) | ||
Theorem | ttukeylem4 10382* | Lemma for ttukey 10388. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ (π β (πΊββ ) = π΅) | ||
Theorem | ttukeylem5 10383* | Lemma for ttukey 10388. The πΊ function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ ((π β§ (πΆ β On β§ π· β On β§ πΆ β π·)) β (πΊβπΆ) β (πΊβπ·)) | ||
Theorem | ttukeylem6 10384* | Lemma for ttukey 10388. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ ((π β§ πΆ β suc (cardβ(βͺ π΄ β π΅))) β (πΊβπΆ) β π΄) | ||
Theorem | ttukeylem7 10385* | Lemma for ttukey 10388. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ (π β βπ₯ β π΄ (π΅ β π₯ β§ βπ¦ β π΄ Β¬ π₯ β π¦)) | ||
Theorem | ttukey2g 10386* | The TeichmΓΌller-Tukey Lemma ttukey 10388 with a slightly stronger conclusion: we can set up the maximal element of π΄ so that it also contains some given π΅ β π΄ as a subset. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ ((βͺ π΄ β dom card β§ π΅ β π΄ β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ (π΅ β π₯ β§ βπ¦ β π΄ Β¬ π₯ β π¦)) | ||
Theorem | ttukeyg 10387* | The TeichmΓΌller-Tukey Lemma ttukey 10388 stated with the "choice" as an antecedent (the hypothesis βͺ π΄ β dom card says that βͺ π΄ is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.) |
β’ ((βͺ π΄ β dom card β§ π΄ β β β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | ttukey 10388* | The TeichmΓΌller-Tukey Lemma, an Axiom of Choice equivalent. If π΄ is a nonempty collection of finite character, then π΄ has a maximal element with respect to inclusion. Here "finite character" means that π₯ β π΄ iff every finite subset of π₯ is in π΄. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ π΄ β V β β’ ((π΄ β β β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | axdclem 10389* | Lemma for axdc 10391. (Contributed by Mario Carneiro, 25-Jan-2013.) |
β’ πΉ = (rec((π¦ β V β¦ (πβ{π§ β£ π¦π₯π§})), π ) βΎ Ο) β β’ ((βπ¦ β π« dom π₯(π¦ β β β (πβπ¦) β π¦) β§ ran π₯ β dom π₯ β§ βπ§(πΉβπΎ)π₯π§) β (πΎ β Ο β (πΉβπΎ)π₯(πΉβsuc πΎ))) | ||
Theorem | axdclem2 10390* | Lemma for axdc 10391. Using the full Axiom of Choice, we can construct a choice function π on π« dom π₯. From this, we can build a sequence πΉ starting at any value π β dom π₯ by repeatedly applying π to the set (πΉβπ₯) (where π₯ is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.) |
β’ πΉ = (rec((π¦ β V β¦ (πβ{π§ β£ π¦π₯π§})), π ) βΎ Ο) β β’ (βπ§ π π₯π§ β (ran π₯ β dom π₯ β βπβπ β Ο (πβπ)π₯(πβsuc π))) | ||
Theorem | axdc 10391* | This theorem derives ax-dc 10316 using ax-ac 10329 and ax-inf 9508. 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.) |
β’ ((βπ¦βπ§ π¦π₯π§ β§ ran π₯ β dom π₯) β βπβπ β Ο (πβπ)π₯(πβsuc π)) | ||
Theorem | fodomg 10392 | An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the axiom of choice ac7g 10344. The axiom of choice is not needed for finite sets, see fodomfi 9203. See also fodomnum 9927. (Contributed by NM, 23-Jul-2004.) (Proof shortened by BJ, 20-May-2024.) |
β’ (π΄ β π β (πΉ:π΄βontoβπ΅ β π΅ βΌ π΄)) | ||
Theorem | fodom 10393 | An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.) |
β’ π΄ β V β β’ (πΉ:π΄βontoβπ΅ β π΅ βΌ π΄) | ||
Theorem | dmct 10394 | The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
β’ (π΄ βΌ Ο β dom π΄ βΌ Ο) | ||
Theorem | rnct 10395 | The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
β’ (π΄ βΌ Ο β ran π΄ βΌ Ο) | ||
Theorem | fodomb 10396* | Equivalence of an onto mapping and dominance for a nonempty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.) |
β’ ((π΄ β β β§ βπ π:π΄βontoβπ΅) β (β βΊ π΅ β§ π΅ βΌ π΄)) | ||
Theorem | wdomac 10397 | 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.) |
β’ (π βΌ* π β π βΌ π) | ||
Theorem | brdom3 10398* | Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.) |
β’ π΅ β V β β’ (π΄ βΌ π΅ β βπ(βπ₯β*π¦ π₯ππ¦ β§ βπ₯ β π΄ βπ¦ β π΅ π¦ππ₯)) | ||
Theorem | brdom5 10399* | An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.) |
β’ π΅ β V β β’ (π΄ βΌ π΅ β βπ(βπ₯ β π΅ β*π¦ π₯ππ¦ β§ βπ₯ β π΄ βπ¦ β π΅ π¦ππ₯)) | ||
Theorem | brdom4 10400* | An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
β’ π΅ β V β β’ (π΄ βΌ π΅ β βπ(βπ₯ β π΅ β*π¦ β π΄ π₯ππ¦ β§ βπ₯ β π΄ βπ¦ β π΅ π¦ππ₯)) |
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