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Type | Label | Description |
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Statement | ||
Theorem | fin1a2lem8 10401* | Lemma for fin1a2 10409. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
β’ ((π΄ β π β§ βπ₯ β π« π΄(π₯ β FinIII β¨ (π΄ β π₯) β FinIII)) β π΄ β FinIII) | ||
Theorem | fin1a2lem9 10402* | Lemma for fin1a2 10409. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
β’ (( [β] Or π β§ π β Fin β§ π΄ β Ο) β {π β π β£ π βΌ π΄} β Fin) | ||
Theorem | fin1a2lem10 10403 | Lemma for fin1a2 10409. A nonempty finite union of members of a chain is a member of the chain. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
β’ ((π΄ β β β§ π΄ β Fin β§ [β] Or π΄) β βͺ π΄ β π΄) | ||
Theorem | fin1a2lem11 10404* | Lemma for fin1a2 10409. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
β’ (( [β] Or π΄ β§ π΄ β Fin) β ran (π β Ο β¦ βͺ {π β π΄ β£ π βΌ π}) = (π΄ βͺ {β })) | ||
Theorem | fin1a2lem12 10405 | Lemma for fin1a2 10409. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
β’ (((π΄ β π« π΅ β§ [β] Or π΄ β§ Β¬ βͺ π΄ β π΄) β§ (π΄ β Fin β§ π΄ β β )) β Β¬ π΅ β FinIII) | ||
Theorem | fin1a2lem13 10406 | Lemma for fin1a2 10409. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
β’ (((π΄ β π« π΅ β§ [β] Or π΄ β§ Β¬ βͺ π΄ β π΄) β§ (Β¬ πΆ β Fin β§ πΆ β π΄)) β Β¬ (π΅ β πΆ) β FinII) | ||
Theorem | fin12 10407 | Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 10409. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
β’ (π΄ β Fin β π΄ β FinII) | ||
Theorem | fin1a2s 10408* | An II-infinite set can have an I-infinite part broken off and remain II-infinite. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.) |
β’ ((π΄ β π β§ βπ₯ β π« π΄(π₯ β Fin β¨ (π΄ β π₯) β FinII)) β π΄ β FinII) | ||
Theorem | fin1a2 10409 | Every Ia-finite set is II-finite. Theorem 1 of [Levy58], p. 3. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.) |
β’ (π΄ β FinIa β π΄ β FinII) | ||
Theorem | itunifval 10410* | Function value of iterated unions. EDITORIAL: The iterated unions and order types of ordered sets are split out here because they could conceivably be independently useful. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) β β’ (π΄ β π β (πβπ΄) = (rec((π¦ β V β¦ βͺ π¦), π΄) βΎ Ο)) | ||
Theorem | itunifn 10411* | Functionality of the iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) β β’ (π΄ β π β (πβπ΄) Fn Ο) | ||
Theorem | ituni0 10412* | A zero-fold iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) β β’ (π΄ β π β ((πβπ΄)ββ ) = π΄) | ||
Theorem | itunisuc 10413* | Successor iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) β β’ ((πβπ΄)βsuc π΅) = βͺ ((πβπ΄)βπ΅) | ||
Theorem | itunitc1 10414* | Each union iterate is a member of the transitive closure. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) β β’ ((πβπ΄)βπ΅) β (TCβπ΄) | ||
Theorem | itunitc 10415* | The union of all union iterates creates the transitive closure; compare trcl 9722. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) β β’ (TCβπ΄) = βͺ ran (πβπ΄) | ||
Theorem | ituniiun 10416* | Unwrap an iterated union from the "other end". (Contributed by Stefan O'Rear, 11-Feb-2015.) |
β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) β β’ (π΄ β π β ((πβπ΄)βsuc π΅) = βͺ π β π΄ ((πβπ)βπ΅)) | ||
Theorem | hsmexlem7 10417* | Lemma for hsmex 10426. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) β β’ (π»ββ ) = (harβπ« π) | ||
Theorem | hsmexlem8 10418* | Lemma for hsmex 10426. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) β β’ (π β Ο β (π»βsuc π) = (harβπ« (π Γ (π»βπ)))) | ||
Theorem | hsmexlem9 10419* | Lemma for hsmex 10426. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) β β’ (π β Ο β (π»βπ) β On) | ||
Theorem | hsmexlem1 10420 | Lemma for hsmex 10426. Bound the order type of a limited-cardinality set of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) |
β’ π = OrdIso( E , π΄) β β’ ((π΄ β On β§ π΄ βΌ* π΅) β dom π β (harβπ« π΅)) | ||
Theorem | hsmexlem2 10421* | Lemma for hsmex 10426. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 10569 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 18-Sep-2021.) |
β’ πΉ = OrdIso( E , π΅) & β’ πΊ = OrdIso( E , βͺ π β π΄ π΅) β β’ ((π΄ β π β§ πΆ β On β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β dom πΊ β (harβπ« (π΄ Γ πΆ))) | ||
Theorem | hsmexlem3 10422* | Lemma for hsmex 10426. Clear πΌ 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.) |
β’ πΉ = OrdIso( E , π΅) & β’ πΊ = OrdIso( E , βͺ π β π΄ π΅) β β’ (((π΄ βΌ* π· β§ πΆ β On) β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β dom πΊ β (harβπ« (π· Γ πΆ))) | ||
Theorem | hsmexlem4 10423* | Lemma for hsmex 10426. The core induction, establishing bounds on the order types of iterated unions of the initial set. (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 Ο β§ π β π) β dom π β (π»βπ)) | ||
Theorem | hsmexlem5 10424* | Lemma for hsmex 10426. Combining the above constraints, along with itunitc 10415 and tcrank 9878, gives an effective constraint on the rank of π. (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 π β (rankβπ) β (harβπ« (Ο Γ βͺ ran π»))) | ||
Theorem | hsmexlem6 10425* | Lemma for hsmex 10426. (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 10426* | 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 9586. (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 10427* | The set of hereditary size-limited sets, assuming ax-reg 9586. (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 10428* | The set of hereditary size-limited sets, assuming ax-reg 9586, 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 10453, as well as weaker forms such as the axiom of countable choice ax-cc 10429 and dependent choice ax-dc 10440. 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 10429* | The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10471, but is weak enough that it can be proven using DC (see axcc 10452). 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 10430* | Lemma for axcc2 10431. (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 10431* | 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 10432* | A possibly more useful version of ax-cc 10429 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 10433* | A version of axcc3 10432 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 10434 | An ax-cc 10429 equivalent: every set has choice sets of length Ο. (Contributed by Mario Carneiro, 31-Aug-2015.) |
β’ AC Ο = V | ||
Theorem | axcc4dom 10435* | Relax the constraint on axcc4 10433 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 10436* | Lemma for domtriom 10437. (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 10437 | Trichotomy of equinumerosity for Ο, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 10308) 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 10438 | 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 10439 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 10429. See dominfac 10567 for a version proved from ax-ac 10453. The axiom of Regularity is used for this proof, via inf3lem6 9627, 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 10440* | 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 10515. 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 10441 | 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 10442* | Lemma for axdc2 10443. 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 10440 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 10443* | An apparent strengthening of ax-dc 10440 (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 10444* | 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 10445* | Lemma for axdc3 10448. 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 10440 (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 10440 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 10446* | Simple substitution lemma for axdc3 10448. (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 10447* | Lemma for axdc3 10448. 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 10440 (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 10440 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 10443. See axdc3lem2 10445 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 10448* | 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 10449* | Lemma for axdc4 10450. (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 10450* | A more general version of axdc3 10448 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 10451* | Lemma for axcc 10452. (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 10452* | Although CC can be proven trivially using ac5 10471, 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 10453* |
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 10456 for a more detailed explanation. Theorem ac2 10455 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 10459 is slightly shorter when the biconditional of ax-ac 10453 is expanded into implication and negation. In axac3 10458 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10675 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 10486, ac5 10471, and ac7 10467. The Axiom of Regularity ax-reg 9586 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as Theorem dfac2b 10124. Equivalents to AC are the well-ordering theorem weth 10489 and Zorn's lemma zorn 10501. See ac4 10469 for comments about stronger versions of AC. In order to avoid uses of ax-reg 9586 for derivation of AC equivalents, we provide ax-ac2 10457 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 10457 from ax-ac 10453 is shown by Theorem axac2 10460, and the reverse derivation by axac 10461. Therefore, new proofs should normally use ax-ac2 10457 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 10454* | Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 10453. (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 10455* | 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 10456 is easier to understand.) Note: aceq0 10112 shows the logical equivalence to ax-ac 10453. (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 10456* |
Axiom of Choice using abbreviations. The logical equivalence to ax-ac 10453
can be established by chaining aceq0 10112 and aceq2 10113. A standard
textbook version of AC is derived from this one in dfac2a 10123, and this
version of AC is derived from the textbook version in dfac2b 10124, showing
their logical equivalence (see dfac2 10125).
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 9612. The key theorem for this (used in the proof of dfac2b 10124) is preleq 9610. 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 10457* | In order to avoid uses of ax-reg 9586 for derivation of AC equivalents, we provide ax-ac2 10457, 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 10459. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1797 available. The derivation of ax-ac2 10457 from ax-ac 10453 is shown by Theorem axac2 10460, and the reverse derivation by axac 10461. Note that we use ax-reg 9586 to derive ax-ac 10453 from ax-ac2 10457, but not to derive ax-ac2 10457 from ax-ac 10453. (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 10458 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10457 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.) |
β’ CHOICE | ||
Theorem | ackm 10459* |
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 10160. 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 10160.
The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html 10160 http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html 10160. (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 10460* | Derive ax-ac2 10457 from ax-ac 10453. (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 10461* | Derive ax-ac 10453 from ax-ac2 10457. Note that ax-reg 9586 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 10462 | Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ (CHOICE β βπ₯π) β β’ π | ||
Theorem | cardeqv 10463 | All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.) |
β’ dom card = V | ||
Theorem | numth3 10464 | 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 10465* | 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 10466* | 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 10467* | 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 10468* | 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 10469* |
Equivalent of Axiom of Choice. We do not insist that π be a
function. However, Theorem ac5 10471, 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 10485. (Contributed by NM, 21-Jul-1996.) |
β’ βπβπ§ β π₯ (π§ β β β (πβπ§) β π§) | ||
Theorem | ac4c 10470* | 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 10471* | 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 10469. (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 10472* | 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 10473* | A version of ac6 10474 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 10474* | 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 10478, 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 10475* | 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 10476* | 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 10477* | 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 10478* | Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9887, we derive this strong version of ac6 10474 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 10479* | Equivalent of Axiom of Choice. Contrapositive of ac6s 10478. (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 10480* | Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 10481. (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 10481* | 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 10482* | ac6s 10478 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 10483* | Version of ac6 10474 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 10484* | 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 10485* | 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 10486* | 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 10487* | 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 9885). (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 10488* | 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 10489* | 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 10490* | Lemma for zorn2 10500. (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 10491* | Lemma for zorn2 10500. (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 10492* | Lemma for zorn2 10500. (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 10493* | Lemma for zorn2 10500. (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 10494* | Lemma for zorn2 10500. (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 10495* | Lemma for zorn2 10500. (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 10496* | Lemma for zorn2 10500. (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 10497* | Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10500 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 10498* | 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 10501 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 10499* | Variant of Zorn's lemma zorng 10498 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 10500* | 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 10490 through zorn2lem7 10496; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 10496. (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¦) |
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