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Type | Label | Description |
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Statement | ||
Theorem | fin1a2lem7 10401* | Lemma for fin1a2 10410. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
β’ πΈ = (π₯ β Ο β¦ (2o Β·o π₯)) & β’ π = (π₯ β On β¦ suc π₯) β β’ ((π΄ β π β§ βπ¦ β π« π΄(π¦ β FinIII β¨ (π΄ β π¦) β FinIII)) β π΄ β FinIII) | ||
Theorem | fin1a2lem8 10402* | Lemma for fin1a2 10410. 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 10403* | Lemma for fin1a2 10410. 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 10404 | Lemma for fin1a2 10410. 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 10405* | Lemma for fin1a2 10410. (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 10406 | Lemma for fin1a2 10410. (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 10407 | Lemma for fin1a2 10410. (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 10408 | Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 10410. (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 10409* | 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 10410 | 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 10411* | 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 10412* | 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 10413* | 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 10414* | 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 10415* | 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 10416* | The union of all union iterates creates the transitive closure; compare trcl 9723. (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 10417* | 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 10418* | Lemma for hsmex 10427. 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 10419* | Lemma for hsmex 10427. 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 10420* | Lemma for hsmex 10427. 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 10421 | Lemma for hsmex 10427. 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 10422* | Lemma for hsmex 10427. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 10570 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 10423* | Lemma for hsmex 10427. 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 10424* | Lemma for hsmex 10427. 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 10425* | Lemma for hsmex 10427. Combining the above constraints, along with itunitc 10416 and tcrank 9879, 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 10426* | Lemma for hsmex 10427. (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 10427* | 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 9587. (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 10428* | The set of hereditary size-limited sets, assuming ax-reg 9587. (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 10429* | The set of hereditary size-limited sets, assuming ax-reg 9587, 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 10454, as well as weaker forms such as the axiom of countable choice ax-cc 10430 and dependent choice ax-dc 10441. 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 10430* | The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10472, but is weak enough that it can be proven using DC (see axcc 10453). 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 10431* | Lemma for axcc2 10432. (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 10432* | 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 10433* | A possibly more useful version of ax-cc 10430 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 10434* | A version of axcc3 10433 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 10435 | An ax-cc 10430 equivalent: every set has choice sets of length Ο. (Contributed by Mario Carneiro, 31-Aug-2015.) |
β’ AC Ο = V | ||
Theorem | axcc4dom 10436* | Relax the constraint on axcc4 10434 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 10437* | Lemma for domtriom 10438. (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 10438 | Trichotomy of equinumerosity for Ο, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 10309) 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 10439 | 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 10440 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 10430. See dominfac 10568 for a version proved from ax-ac 10454. The axiom of Regularity is used for this proof, via inf3lem6 9628, 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 10441* | 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 10516. 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 10442 | 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 10443* | Lemma for axdc2 10444. 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 10441 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 10444* | An apparent strengthening of ax-dc 10441 (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 10445* | 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 10446* | Lemma for axdc3 10449. 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 10441 (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 10441 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 10447* | Simple substitution lemma for axdc3 10449. (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 10448* | Lemma for axdc3 10449. 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 10441 (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 10441 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 10444. See axdc3lem2 10446 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 10449* | 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 10450* | Lemma for axdc4 10451. (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 10451* | A more general version of axdc3 10449 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 10452* | Lemma for axcc 10453. (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 10453* | Although CC can be proven trivially using ac5 10472, 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 10454* |
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 10457 for a more detailed explanation. Theorem ac2 10456 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 10460 is slightly shorter when the biconditional of ax-ac 10454 is expanded into implication and negation. In axac3 10459 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10676 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 10487, ac5 10472, and ac7 10468. The Axiom of Regularity ax-reg 9587 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as Theorem dfac2b 10125. Equivalents to AC are the well-ordering theorem weth 10490 and Zorn's lemma zorn 10502. See ac4 10470 for comments about stronger versions of AC. In order to avoid uses of ax-reg 9587 for derivation of AC equivalents, we provide ax-ac2 10458 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 10458 from ax-ac 10454 is shown by Theorem axac2 10461, and the reverse derivation by axac 10462. Therefore, new proofs should normally use ax-ac2 10458 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 10455* | Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 10454. (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 10456* | 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 10457 is easier to understand.) Note: aceq0 10113 shows the logical equivalence to ax-ac 10454. (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 10457* |
Axiom of Choice using abbreviations. The logical equivalence to ax-ac 10454
can be established by chaining aceq0 10113 and aceq2 10114. A standard
textbook version of AC is derived from this one in dfac2a 10124, and this
version of AC is derived from the textbook version in dfac2b 10125, showing
their logical equivalence (see dfac2 10126).
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 9613. The key theorem for this (used in the proof of dfac2b 10125) is preleq 9611. 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 10458* | In order to avoid uses of ax-reg 9587 for derivation of AC equivalents, we provide ax-ac2 10458, 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 10460. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1798 available. The derivation of ax-ac2 10458 from ax-ac 10454 is shown by Theorem axac2 10461, and the reverse derivation by axac 10462. Note that we use ax-reg 9587 to derive ax-ac 10454 from ax-ac2 10458, but not to derive ax-ac2 10458 from ax-ac 10454. (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 10459 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10458 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.) |
β’ CHOICE | ||
Theorem | ackm 10460* |
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 10161. 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 10161.
The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html 10161 http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html 10161. (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 10461* | Derive ax-ac2 10458 from ax-ac 10454. (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 10462* | Derive ax-ac 10454 from ax-ac2 10458. Note that ax-reg 9587 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 10463 | Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ (CHOICE β βπ₯π) β β’ π | ||
Theorem | cardeqv 10464 | All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.) |
β’ dom card = V | ||
Theorem | numth3 10465 | 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 10466* | 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 10467* | 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 10468* | 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 10469* | 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 10470* |
Equivalent of Axiom of Choice. We do not insist that π be a
function. However, Theorem ac5 10472, 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 10486. (Contributed by NM, 21-Jul-1996.) |
β’ βπβπ§ β π₯ (π§ β β β (πβπ§) β π§) | ||
Theorem | ac4c 10471* | 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 10472* | 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 10470. (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 10473* | 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 10474* | A version of ac6 10475 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 10475* | 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 10479, 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 10476* | 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 10477* | 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 10478* | 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 10479* | Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9888, we derive this strong version of ac6 10475 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 10480* | Equivalent of Axiom of Choice. Contrapositive of ac6s 10479. (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 10481* | Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 10482. (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 10482* | 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 10483* | ac6s 10479 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 10484* | Version of ac6 10475 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 10485* | 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 10486* | 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 10487* | 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 10488* | 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 9886). (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 10489* | 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 10490* | 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 10491* | Lemma for zorn2 10501. (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 10492* | Lemma for zorn2 10501. (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 10493* | Lemma for zorn2 10501. (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 10494* | Lemma for zorn2 10501. (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 10495* | Lemma for zorn2 10501. (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 10496* | Lemma for zorn2 10501. (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 10497* | Lemma for zorn2 10501. (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 10498* | Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10501 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 10499* | 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 10502 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 10500* | Variant of Zorn's lemma zorng 10499 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¦) |
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