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Theorem List for Metamath Proof Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhsmexlem6 10301* Lemma for hsmex 10302. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝑋 ∈ V    &   π» = (rec((𝑧 ∈ V ↦ (harβ€˜π’« (𝑋 Γ— 𝑧))), (harβ€˜π’« 𝑋)) β†Ύ Ο‰)    &   π‘ˆ = (π‘₯ ∈ V ↦ (rec((𝑦 ∈ V ↦ βˆͺ 𝑦), π‘₯) β†Ύ Ο‰))    &   π‘† = {π‘Ž ∈ βˆͺ (𝑅1 β€œ On) ∣ βˆ€π‘ ∈ (TCβ€˜{π‘Ž})𝑏 β‰Ό 𝑋}    &   π‘‚ = OrdIso( E , (rank β€œ ((π‘ˆβ€˜π‘‘)β€˜π‘)))    β‡’   π‘† ∈ V
 
Theoremhsmex 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.)
(𝑋 ∈ 𝑉 β†’ {𝑠 ∈ βˆͺ (𝑅1 β€œ On) ∣ βˆ€π‘₯ ∈ (TCβ€˜{𝑠})π‘₯ β‰Ό 𝑋} ∈ V)
 
Theoremhsmex2 10303* The set of hereditary size-limited sets, assuming ax-reg 9462. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋 ∈ 𝑉 β†’ {𝑠 ∣ βˆ€π‘₯ ∈ (TCβ€˜{𝑠})π‘₯ β‰Ό 𝑋} ∈ V)
 
Theoremhsmex3 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.)
(𝑋 ∈ 𝑉 β†’ {𝑠 ∣ βˆ€π‘₯ ∈ (TCβ€˜{𝑠})π‘₯ β‰Ί 𝑋} ∈ V)
 
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY

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.

 
3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
 
3.1.1  Introduce the Axiom of Countable Choice
 
Axiomax-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.)
(π‘₯ β‰ˆ Ο‰ β†’ βˆƒπ‘“βˆ€π‘§ ∈ π‘₯ (𝑧 β‰  βˆ… β†’ (π‘“β€˜π‘§) ∈ 𝑧))
 
Theoremaxcc2lem 10306* Lemma for axcc2 10307. (Contributed by Mario Carneiro, 8-Feb-2013.)
𝐾 = (𝑛 ∈ Ο‰ ↦ if((πΉβ€˜π‘›) = βˆ…, {βˆ…}, (πΉβ€˜π‘›)))    &   π΄ = (𝑛 ∈ Ο‰ ↦ ({𝑛} Γ— (πΎβ€˜π‘›)))    &   πΊ = (𝑛 ∈ Ο‰ ↦ (2nd β€˜(π‘“β€˜(π΄β€˜π‘›))))    β‡’   βˆƒπ‘”(𝑔 Fn Ο‰ ∧ βˆ€π‘› ∈ Ο‰ ((πΉβ€˜π‘›) β‰  βˆ… β†’ (π‘”β€˜π‘›) ∈ (πΉβ€˜π‘›)))
 
Theoremaxcc2 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.)
βˆƒπ‘”(𝑔 Fn Ο‰ ∧ βˆ€π‘› ∈ Ο‰ ((πΉβ€˜π‘›) β‰  βˆ… β†’ (π‘”β€˜π‘›) ∈ (πΉβ€˜π‘›)))
 
Theoremaxcc3 10308* A possibly more useful version of ax-cc 10305 using sequences 𝐹(𝑛) 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.)
𝐹 ∈ V    &   π‘ β‰ˆ Ο‰    β‡’   βˆƒπ‘“(𝑓 Fn 𝑁 ∧ βˆ€π‘› ∈ 𝑁 (𝐹 β‰  βˆ… β†’ (π‘“β€˜π‘›) ∈ 𝐹))
 
Theoremaxcc4 10309* A version of axcc3 10308 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.)
𝐴 ∈ V    &   π‘ β‰ˆ Ο‰    &   (π‘₯ = (π‘“β€˜π‘›) β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘› ∈ 𝑁 βˆƒπ‘₯ ∈ 𝐴 πœ‘ β†’ βˆƒπ‘“(𝑓:π‘βŸΆπ΄ ∧ βˆ€π‘› ∈ 𝑁 πœ“))
 
Theoremacncc 10310 An ax-cc 10305 equivalent: every set has choice sets of length Ο‰. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC Ο‰ = V
 
Theoremaxcc4dom 10311* Relax the constraint on axcc4 10309 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.)
𝐴 ∈ V    &   (π‘₯ = (π‘“β€˜π‘›) β†’ (πœ‘ ↔ πœ“))    β‡’   ((𝑁 β‰Ό Ο‰ ∧ βˆ€π‘› ∈ 𝑁 βˆƒπ‘₯ ∈ 𝐴 πœ‘) β†’ βˆƒπ‘“(𝑓:π‘βŸΆπ΄ ∧ βˆ€π‘› ∈ 𝑁 πœ“))
 
Theoremdomtriomlem 10312* Lemma for domtriom 10313. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V    &   π΅ = {𝑦 ∣ (𝑦 βŠ† 𝐴 ∧ 𝑦 β‰ˆ 𝒫 𝑛)}    &   πΆ = (𝑛 ∈ Ο‰ ↦ ((π‘β€˜π‘›) βˆ– βˆͺ π‘˜ ∈ 𝑛 (π‘β€˜π‘˜)))    β‡’   (Β¬ 𝐴 ∈ Fin β†’ Ο‰ β‰Ό 𝐴)
 
Theoremdomtriom 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.)
𝐴 ∈ V    β‡’   (Ο‰ β‰Ό 𝐴 ↔ Β¬ 𝐴 β‰Ί Ο‰)
 
Theoremfin41 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.)
FinIV = Fin
 
Theoremdominf 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 𝐴 = {𝐴} or 𝐴 = {βˆ…, 𝐴} (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V    β‡’   ((𝐴 β‰  βˆ… ∧ 𝐴 βŠ† βˆͺ 𝐴) β†’ Ο‰ β‰Ό 𝐴)
 
3.1.2  Introduce the Axiom of Dependent Choice
 
Axiomax-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.)
((βˆƒπ‘¦βˆƒπ‘§ 𝑦π‘₯𝑧 ∧ ran π‘₯ βŠ† dom π‘₯) β†’ βˆƒπ‘“βˆ€π‘› ∈ Ο‰ (π‘“β€˜π‘›)π‘₯(π‘“β€˜suc 𝑛))
 
Theoremdcomex 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.)
Ο‰ ∈ V
 
Theoremaxdc2lem 10318* Lemma for axdc2 10319. We construct a relation 𝑅 based on 𝐹 such that π‘₯𝑅𝑦 iff 𝑦 ∈ (πΉβ€˜π‘₯), 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.)
𝐴 ∈ V    &   π‘… = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (πΉβ€˜π‘₯))}    &   πΊ = (π‘₯ ∈ Ο‰ ↦ (β„Žβ€˜π‘₯))    β‡’   ((𝐴 β‰  βˆ… ∧ 𝐹:𝐴⟢(𝒫 𝐴 βˆ– {βˆ…})) β†’ βˆƒπ‘”(𝑔:Ο‰βŸΆπ΄ ∧ βˆ€π‘˜ ∈ Ο‰ (π‘”β€˜suc π‘˜) ∈ (πΉβ€˜(π‘”β€˜π‘˜))))
 
Theoremaxdc2 10319* An apparent strengthening of ax-dc 10316 (but derived from it) which shows that there is a denumerable sequence 𝑔 for any function that maps elements of a set 𝐴 to nonempty subsets of 𝐴 such that 𝑔(π‘₯ + 1) ∈ 𝐹(𝑔(π‘₯)) for all π‘₯ ∈ 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.)
𝐴 ∈ V    β‡’   ((𝐴 β‰  βˆ… ∧ 𝐹:𝐴⟢(𝒫 𝐴 βˆ– {βˆ…})) β†’ βˆƒπ‘”(𝑔:Ο‰βŸΆπ΄ ∧ βˆ€π‘˜ ∈ Ο‰ (π‘”β€˜suc π‘˜) ∈ (πΉβ€˜(π‘”β€˜π‘˜))))
 
Theoremaxdc3lem 10320* The class 𝑆 of finite approximations to the DC sequence is a set. (We derive here the stronger statement that 𝑆 is a subset of a specific set, namely 𝒫 (Ο‰ Γ— 𝐴).) (Contributed by Mario Carneiro, 27-Jan-2013.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 18-Mar-2014.)
𝐴 ∈ V    &   π‘† = {𝑠 ∣ βˆƒπ‘› ∈ Ο‰ (𝑠:suc π‘›βŸΆπ΄ ∧ (π‘ β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ 𝑛 (π‘ β€˜suc π‘˜) ∈ (πΉβ€˜(π‘ β€˜π‘˜)))}    β‡’   π‘† ∈ V
 
Theoremaxdc3lem2 10321* Lemma for axdc3 10324. We have constructed a "candidate set" 𝑆, which consists of all finite sequences 𝑠 that satisfy our property of interest, namely 𝑠(π‘₯ + 1) ∈ 𝐹(𝑠(π‘₯)) on its domain, but with the added constraint that 𝑠(0) = 𝐢. 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 (β„Žβ€˜π‘›):π‘šβŸΆπ΄ (for some integer π‘š). 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„Ž, we can construct the sequence 𝑔 that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.)
𝐴 ∈ V    &   π‘† = {𝑠 ∣ βˆƒπ‘› ∈ Ο‰ (𝑠:suc π‘›βŸΆπ΄ ∧ (π‘ β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ 𝑛 (π‘ β€˜suc π‘˜) ∈ (πΉβ€˜(π‘ β€˜π‘˜)))}    &   πΊ = (π‘₯ ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ (dom 𝑦 = suc dom π‘₯ ∧ (𝑦 β†Ύ dom π‘₯) = π‘₯)})    β‡’   (βˆƒβ„Ž(β„Ž:Ο‰βŸΆπ‘† ∧ βˆ€π‘˜ ∈ Ο‰ (β„Žβ€˜suc π‘˜) ∈ (πΊβ€˜(β„Žβ€˜π‘˜))) β†’ βˆƒπ‘”(𝑔:Ο‰βŸΆπ΄ ∧ (π‘”β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ Ο‰ (π‘”β€˜suc π‘˜) ∈ (πΉβ€˜(π‘”β€˜π‘˜))))
 
Theoremaxdc3lem3 10322* Simple substitution lemma for axdc3 10324. (Contributed by Mario Carneiro, 27-Jan-2013.)
𝐴 ∈ V    &   π‘† = {𝑠 ∣ βˆƒπ‘› ∈ Ο‰ (𝑠:suc π‘›βŸΆπ΄ ∧ (π‘ β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ 𝑛 (π‘ β€˜suc π‘˜) ∈ (πΉβ€˜(π‘ β€˜π‘˜)))}    &   π΅ ∈ V    β‡’   (𝐡 ∈ 𝑆 ↔ βˆƒπ‘š ∈ Ο‰ (𝐡:suc π‘šβŸΆπ΄ ∧ (π΅β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ π‘š (π΅β€˜suc π‘˜) ∈ (πΉβ€˜(π΅β€˜π‘˜))))
 
Theoremaxdc3lem4 10323* Lemma for axdc3 10324. We have constructed a "candidate set" 𝑆, which consists of all finite sequences 𝑠 that satisfy our property of interest, namely 𝑠(π‘₯ + 1) ∈ 𝐹(𝑠(π‘₯)) on its domain, but with the added constraint that 𝑠(0) = 𝐢. 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 (β„Žβ€˜π‘›):π‘šβŸΆπ΄ (for some integer π‘š). 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 𝑆 is nonempty, and that 𝐺 always maps to a nonempty subset of 𝑆, so that we can apply axdc2 10319. See axdc3lem2 10321 for the rest of the proof. (Contributed by Mario Carneiro, 27-Jan-2013.)
𝐴 ∈ V    &   π‘† = {𝑠 ∣ βˆƒπ‘› ∈ Ο‰ (𝑠:suc π‘›βŸΆπ΄ ∧ (π‘ β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ 𝑛 (π‘ β€˜suc π‘˜) ∈ (πΉβ€˜(π‘ β€˜π‘˜)))}    &   πΊ = (π‘₯ ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ (dom 𝑦 = suc dom π‘₯ ∧ (𝑦 β†Ύ dom π‘₯) = π‘₯)})    β‡’   ((𝐢 ∈ 𝐴 ∧ 𝐹:𝐴⟢(𝒫 𝐴 βˆ– {βˆ…})) β†’ βˆƒπ‘”(𝑔:Ο‰βŸΆπ΄ ∧ (π‘”β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ Ο‰ (π‘”β€˜suc π‘˜) ∈ (πΉβ€˜(π‘”β€˜π‘˜))))
 
Theoremaxdc3 10324* Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value 𝐢. 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.)
𝐴 ∈ V    β‡’   ((𝐢 ∈ 𝐴 ∧ 𝐹:𝐴⟢(𝒫 𝐴 βˆ– {βˆ…})) β†’ βˆƒπ‘”(𝑔:Ο‰βŸΆπ΄ ∧ (π‘”β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ Ο‰ (π‘”β€˜suc π‘˜) ∈ (πΉβ€˜(π‘”β€˜π‘˜))))
 
Theoremaxdc4lem 10325* Lemma for axdc4 10326. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
𝐴 ∈ V    &   πΊ = (𝑛 ∈ Ο‰, π‘₯ ∈ 𝐴 ↦ ({suc 𝑛} Γ— (𝑛𝐹π‘₯)))    β‡’   ((𝐢 ∈ 𝐴 ∧ 𝐹:(Ο‰ Γ— 𝐴)⟢(𝒫 𝐴 βˆ– {βˆ…})) β†’ βˆƒπ‘”(𝑔:Ο‰βŸΆπ΄ ∧ (π‘”β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ Ο‰ (π‘”β€˜suc π‘˜) ∈ (π‘˜πΉ(π‘”β€˜π‘˜))))
 
Theoremaxdc4 10326* A more general version of axdc3 10324 that allows the function 𝐹 to vary with π‘˜. (Contributed by Mario Carneiro, 31-Jan-2013.)
𝐴 ∈ V    β‡’   ((𝐢 ∈ 𝐴 ∧ 𝐹:(Ο‰ Γ— 𝐴)⟢(𝒫 𝐴 βˆ– {βˆ…})) β†’ βˆƒπ‘”(𝑔:Ο‰βŸΆπ΄ ∧ (π‘”β€˜βˆ…) = 𝐢 ∧ βˆ€π‘˜ ∈ Ο‰ (π‘”β€˜suc π‘˜) ∈ (π‘˜πΉ(π‘”β€˜π‘˜))))
 
Theoremaxcclem 10327* Lemma for axcc 10328. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
𝐴 = (π‘₯ βˆ– {βˆ…})    &   πΉ = (𝑛 ∈ Ο‰, 𝑦 ∈ βˆͺ 𝐴 ↦ (π‘“β€˜π‘›))    &   πΊ = (𝑀 ∈ 𝐴 ↦ (β„Žβ€˜suc (β—‘π‘“β€˜π‘€)))    β‡’   (π‘₯ β‰ˆ Ο‰ β†’ βˆƒπ‘”βˆ€π‘§ ∈ π‘₯ (𝑧 β‰  βˆ… β†’ (π‘”β€˜π‘§) ∈ 𝑧))
 
Theoremaxcc 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.)
(π‘₯ β‰ˆ Ο‰ β†’ βˆƒπ‘“βˆ€π‘§ ∈ π‘₯ (𝑧 β‰  βˆ… β†’ (π‘“β€˜π‘§) ∈ 𝑧))
 
3.2  ZFC Set Theory - add the Axiom of Choice
 
3.2.1  Introduce the Axiom of Choice
 
Axiomax-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 𝑦 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.)

βˆƒπ‘¦βˆ€π‘§βˆ€π‘€((𝑧 ∈ 𝑀 ∧ 𝑀 ∈ π‘₯) β†’ βˆƒπ‘£βˆ€π‘’(βˆƒπ‘‘((𝑒 ∈ 𝑀 ∧ 𝑀 ∈ 𝑑) ∧ (𝑒 ∈ 𝑑 ∧ 𝑑 ∈ 𝑦)) ↔ 𝑒 = 𝑣))
 
Theoremzfac 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.)
βˆƒπ‘₯βˆ€π‘¦βˆ€π‘§((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑀) β†’ βˆƒπ‘€βˆ€π‘¦(βˆƒπ‘€((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑀) ∧ (𝑦 ∈ 𝑀 ∧ 𝑀 ∈ π‘₯)) ↔ 𝑦 = 𝑀))
 
Theoremac2 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.)
βˆƒπ‘¦βˆ€π‘§ ∈ π‘₯ βˆ€π‘€ ∈ 𝑧 βˆƒ!𝑣 ∈ 𝑧 βˆƒπ‘’ ∈ 𝑦 (𝑧 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)
 
Theoremac3 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 𝑦 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 𝑦 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 𝑦 is actually a choice function on the members of π‘₯.

For example, suppose π‘₯ = {{1, 2}, {1, 3}, {2, 3, 4}}. Let us try 𝑦 = {{{1, 2}, 1}, {{1, 3}, 1}, {{2, 3, 4}, 2}}. For the member (of π‘₯) 𝑧 = {1, 2}, the only assignment to 𝑀 and 𝑣 that satisfies the axiom is 𝑀 = 1 and 𝑣 = {{1, 2}, 1}, so there is exactly one 𝑀 as required. We verify the other two members of π‘₯ similarly. Thus, 𝑦 satisfies the axiom. Using our modified ordered pair definition, we can say that 𝑦 corresponds to the choice function {⟨{1, 2}, 1⟩, ⟨{1, 3}, 1⟩, ⟨{2, 3, 4}, 2⟩}. Of course other choices for 𝑦 will also satisfy the axiom, for example 𝑦 = {{{1, 2}, 2}, {{1, 3}, 1}, {{2, 3, 4}, 4}}. What AC tells us is that there exists at least one such 𝑦, but it doesn't tell us which one.

(New usage is discouraged.) (Contributed by NM, 19-Jul-1996.)

βˆƒπ‘¦βˆ€π‘§ ∈ π‘₯ (𝑧 β‰  βˆ… β†’ βˆƒ!𝑀 ∈ 𝑧 βˆƒπ‘£ ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑀 ∈ 𝑣))
 
Axiomax-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.)
βˆƒπ‘¦βˆ€π‘§βˆƒπ‘£βˆ€π‘’((𝑦 ∈ π‘₯ ∧ (𝑧 ∈ 𝑦 β†’ ((𝑣 ∈ π‘₯ ∧ Β¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (Β¬ 𝑦 ∈ π‘₯ ∧ (𝑧 ∈ π‘₯ β†’ ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑒 ∈ 𝑧 ∧ 𝑒 ∈ 𝑦) β†’ 𝑒 = 𝑣)))))
 
Theoremaxac3 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.)
CHOICE
 
Theoremackm 10335* A remarkable equivalent to the Axiom of Choice that has only five quantifiers (when expanded to use only the primitive predicates = and ∈ 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 (∈ , =)-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.)

βˆ€π‘₯βˆƒπ‘¦βˆ€π‘§βˆƒπ‘£βˆ€π‘’((𝑦 ∈ π‘₯ ∧ (𝑧 ∈ 𝑦 β†’ ((𝑣 ∈ π‘₯ ∧ Β¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (Β¬ 𝑦 ∈ π‘₯ ∧ (𝑧 ∈ π‘₯ β†’ ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑒 ∈ 𝑧 ∧ 𝑒 ∈ 𝑦) β†’ 𝑒 = 𝑣)))))
 
Theoremaxac2 10336* Derive ax-ac2 10333 from ax-ac 10329. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
βˆƒπ‘¦βˆ€π‘§βˆƒπ‘£βˆ€π‘’((𝑦 ∈ π‘₯ ∧ (𝑧 ∈ 𝑦 β†’ ((𝑣 ∈ π‘₯ ∧ Β¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (Β¬ 𝑦 ∈ π‘₯ ∧ (𝑧 ∈ π‘₯ β†’ ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑒 ∈ 𝑧 ∧ 𝑒 ∈ 𝑦) β†’ 𝑒 = 𝑣)))))
 
Theoremaxac 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.)
βˆƒπ‘¦βˆ€π‘§βˆ€π‘€((𝑧 ∈ 𝑀 ∧ 𝑀 ∈ π‘₯) β†’ βˆƒπ‘£βˆ€π‘’(βˆƒπ‘‘((𝑒 ∈ 𝑀 ∧ 𝑀 ∈ 𝑑) ∧ (𝑒 ∈ 𝑑 ∧ 𝑑 ∈ 𝑦)) ↔ 𝑒 = 𝑣))
 
Theoremaxaci 10338 Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ βˆ€π‘₯πœ‘)    β‡’   πœ‘
 
Theoremcardeqv 10339 All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
dom card = V
 
Theoremnumth3 10340 All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
(𝐴 ∈ 𝑉 β†’ 𝐴 ∈ dom card)
 
Theoremnumth2 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.)
𝐴 ∈ V    β‡’   βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴
 
Theoremnumth 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.)
𝐴 ∈ V    β‡’   βˆƒπ‘₯ ∈ On βˆƒπ‘“ 𝑓:π‘₯–1-1-onto→𝐴
 
Theoremac7 10343* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.)
βˆƒπ‘“(𝑓 βŠ† π‘₯ ∧ 𝑓 Fn dom π‘₯)
 
Theoremac7g 10344* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)
(𝑅 ∈ 𝐴 β†’ βˆƒπ‘“(𝑓 βŠ† 𝑅 ∧ 𝑓 Fn dom 𝑅))
 
Theoremac4 10345* Equivalent of Axiom of Choice. We do not insist that 𝑓 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 𝑓 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" βˆƒπΉβˆ€π‘§(𝑧 β‰  βˆ… β†’ (πΉβ€˜π‘§) ∈ 𝑧), which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable 𝐹 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.)

βˆƒπ‘“βˆ€π‘§ ∈ π‘₯ (𝑧 β‰  βˆ… β†’ (π‘“β€˜π‘§) ∈ 𝑧)
 
Theoremac4c 10346* Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.)
𝐴 ∈ V    β‡’   βˆƒπ‘“βˆ€π‘₯ ∈ 𝐴 (π‘₯ β‰  βˆ… β†’ (π‘“β€˜π‘₯) ∈ π‘₯)
 
Theoremac5 10347* An Axiom of Choice equivalent: there exists a function 𝑓 (called a choice function) with domain 𝐴 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 𝑓 be a function is not necessary; see ac4 10345. (Contributed by NM, 29-Aug-1999.)
𝐴 ∈ V    β‡’   βˆƒπ‘“(𝑓 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘₯ β‰  βˆ… β†’ (π‘“β€˜π‘₯) ∈ π‘₯))
 
Theoremac5b 10348* Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)
𝐴 ∈ V    β‡’   (βˆ€π‘₯ ∈ 𝐴 π‘₯ β‰  βˆ… β†’ βˆƒπ‘“(𝑓:𝐴⟢βˆͺ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) ∈ π‘₯))
 
Theoremac6num 10349* A version of ac6 10350 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   ((𝐴 ∈ 𝑉 ∧ βˆͺ π‘₯ ∈ 𝐴 {𝑦 ∈ 𝐡 ∣ πœ‘} ∈ dom card ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) β†’ βˆƒπ‘“(𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 πœ“))
 
Theoremac6 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 𝐡, where πœ‘ depends on π‘₯ (the natural number) and 𝑦 (to specify a member of 𝐡). A stronger version of this theorem, ac6s 10354, allows 𝐡 to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)
𝐴 ∈ V    &   π΅ ∈ V    &   (𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘ β†’ βˆƒπ‘“(𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 πœ“))
 
Theoremac6c4 10351* Equivalent of Axiom of Choice. 𝐡 is a collection 𝐡(π‘₯) of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   π΅ ∈ V    β‡’   (βˆ€π‘₯ ∈ 𝐴 𝐡 β‰  βˆ… β†’ βˆƒπ‘“(𝑓 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) ∈ 𝐡))
 
Theoremac6c5 10352* Equivalent of Axiom of Choice. 𝐡 is a collection 𝐡(π‘₯) of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   π΅ ∈ V    β‡’   (βˆ€π‘₯ ∈ 𝐴 𝐡 β‰  βˆ… β†’ βˆƒπ‘“βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) ∈ 𝐡)
 
Theoremac9 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.)
𝐴 ∈ V    &   π΅ ∈ V    β‡’   (βˆ€π‘₯ ∈ 𝐴 𝐡 β‰  βˆ… ↔ Xπ‘₯ ∈ 𝐴 𝐡 β‰  βˆ…)
 
Theoremac6s 10354* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9763, we derive this strong version of ac6 10350 that doesn't require 𝐡 to be a set. (Contributed by NM, 4-Feb-2004.)
𝐴 ∈ V    &   (𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘ β†’ βˆƒπ‘“(𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 πœ“))
 
Theoremac6n 10355* Equivalent of Axiom of Choice. Contrapositive of ac6s 10354. (Contributed by NM, 10-Jun-2007.)
𝐴 ∈ V    &   (𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘“(𝑓:𝐴⟢𝐡 β†’ βˆƒπ‘₯ ∈ 𝐴 πœ“) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 πœ‘)
 
Theoremac6s2 10356* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 10357. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V    &   (𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦πœ‘ β†’ βˆƒπ‘“(𝑓 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 πœ“))
 
Theoremac6s3 10357* Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)
𝐴 ∈ V    &   (𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦πœ‘ β†’ βˆƒπ‘“βˆ€π‘₯ ∈ 𝐴 πœ“)
 
Theoremac6sg 10358* ac6s 10354 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)
(𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   (𝐴 ∈ 𝑉 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘ β†’ βˆƒπ‘“(𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 πœ“)))
 
Theoremac6sf 10359* Version of ac6 10350 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)
β„²π‘¦πœ“    &   π΄ ∈ V    &   (𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘ β†’ βˆƒπ‘“(𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 πœ“))
 
Theoremac6s4 10360* Generalization of the Axiom of Choice to proper classes. 𝐡 is a collection 𝐡(π‘₯) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V    β‡’   (βˆ€π‘₯ ∈ 𝐴 𝐡 β‰  βˆ… β†’ βˆƒπ‘“(𝑓 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) ∈ 𝐡))
 
Theoremac6s5 10361* Generalization of the Axiom of Choice to proper classes. 𝐡 is a collection 𝐡(π‘₯) of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.)
𝐴 ∈ V    β‡’   (βˆ€π‘₯ ∈ 𝐴 𝐡 β‰  βˆ… β†’ βˆƒπ‘“βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) ∈ 𝐡)
 
Theoremac8 10362* An Axiom of Choice equivalent. Given a family π‘₯ of mutually disjoint nonempty sets, there exists a set 𝑦 containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.)
((βˆ€π‘§ ∈ π‘₯ 𝑧 β‰  βˆ… ∧ βˆ€π‘§ ∈ π‘₯ βˆ€π‘€ ∈ π‘₯ (𝑧 β‰  𝑀 β†’ (𝑧 ∩ 𝑀) = βˆ…)) β†’ βˆƒπ‘¦βˆ€π‘§ ∈ π‘₯ βˆƒ!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
 
Theoremac9s 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 𝐡(π‘₯) (achieved via the Collection Principle cp 9761). (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V    β‡’   (βˆ€π‘₯ ∈ 𝐴 𝐡 β‰  βˆ… ↔ Xπ‘₯ ∈ 𝐴 𝐡 β‰  βˆ…)
 
3.2.2  AC equivalents: well-ordering, Zorn's lemma
 
Theoremnumthcor 10364* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
(𝐴 ∈ 𝑉 β†’ βˆƒπ‘₯ ∈ On 𝐴 β‰Ί π‘₯)
 
Theoremweth 10365* Well-ordering theorem: any set 𝐴 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.)
(𝐴 ∈ 𝑉 β†’ βˆƒπ‘₯ π‘₯ We 𝐴)
 
Theoremzorn2lem1 10366* Lemma for zorn2 10376. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐢 βˆ€π‘’ ∈ 𝐢 Β¬ 𝑒𝑀𝑣)))    &   πΆ = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ ran 𝑓 𝑔𝑅𝑧}    &   π· = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ π‘₯)𝑔𝑅𝑧}    β‡’   ((π‘₯ ∈ On ∧ (𝑀 We 𝐴 ∧ 𝐷 β‰  βˆ…)) β†’ (πΉβ€˜π‘₯) ∈ 𝐷)
 
Theoremzorn2lem2 10367* Lemma for zorn2 10376. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐢 βˆ€π‘’ ∈ 𝐢 Β¬ 𝑒𝑀𝑣)))    &   πΆ = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ ran 𝑓 𝑔𝑅𝑧}    &   π· = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ π‘₯)𝑔𝑅𝑧}    β‡’   ((π‘₯ ∈ On ∧ (𝑀 We 𝐴 ∧ 𝐷 β‰  βˆ…)) β†’ (𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦)𝑅(πΉβ€˜π‘₯)))
 
Theoremzorn2lem3 10368* Lemma for zorn2 10376. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐢 βˆ€π‘’ ∈ 𝐢 Β¬ 𝑒𝑀𝑣)))    &   πΆ = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ ran 𝑓 𝑔𝑅𝑧}    &   π· = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ π‘₯)𝑔𝑅𝑧}    β‡’   ((𝑅 Po 𝐴 ∧ (π‘₯ ∈ On ∧ (𝑀 We 𝐴 ∧ 𝐷 β‰  βˆ…))) β†’ (𝑦 ∈ π‘₯ β†’ Β¬ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦)))
 
Theoremzorn2lem4 10369* Lemma for zorn2 10376. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐢 βˆ€π‘’ ∈ 𝐢 Β¬ 𝑒𝑀𝑣)))    &   πΆ = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ ran 𝑓 𝑔𝑅𝑧}    &   π· = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ π‘₯)𝑔𝑅𝑧}    β‡’   ((𝑅 Po 𝐴 ∧ 𝑀 We 𝐴) β†’ βˆƒπ‘₯ ∈ On 𝐷 = βˆ…)
 
Theoremzorn2lem5 10370* Lemma for zorn2 10376. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐢 βˆ€π‘’ ∈ 𝐢 Β¬ 𝑒𝑀𝑣)))    &   πΆ = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ ran 𝑓 𝑔𝑅𝑧}    &   π· = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ π‘₯)𝑔𝑅𝑧}    &   π» = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ 𝑦)𝑔𝑅𝑧}    β‡’   (((𝑀 We 𝐴 ∧ π‘₯ ∈ On) ∧ βˆ€π‘¦ ∈ π‘₯ 𝐻 β‰  βˆ…) β†’ (𝐹 β€œ π‘₯) βŠ† 𝐴)
 
Theoremzorn2lem6 10371* Lemma for zorn2 10376. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐢 βˆ€π‘’ ∈ 𝐢 Β¬ 𝑒𝑀𝑣)))    &   πΆ = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ ran 𝑓 𝑔𝑅𝑧}    &   π· = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ π‘₯)𝑔𝑅𝑧}    &   π» = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ 𝑦)𝑔𝑅𝑧}    β‡’   (𝑅 Po 𝐴 β†’ (((𝑀 We 𝐴 ∧ π‘₯ ∈ On) ∧ βˆ€π‘¦ ∈ π‘₯ 𝐻 β‰  βˆ…) β†’ 𝑅 Or (𝐹 β€œ π‘₯)))
 
Theoremzorn2lem7 10372* Lemma for zorn2 10376. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐢 βˆ€π‘’ ∈ 𝐢 Β¬ 𝑒𝑀𝑣)))    &   πΆ = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ ran 𝑓 𝑔𝑅𝑧}    &   π· = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ π‘₯)𝑔𝑅𝑧}    &   π» = {𝑧 ∈ 𝐴 ∣ βˆ€π‘” ∈ (𝐹 β€œ 𝑦)𝑔𝑅𝑧}    β‡’   ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ βˆ€π‘ ((𝑠 βŠ† 𝐴 ∧ 𝑅 Or 𝑠) β†’ βˆƒπ‘Ž ∈ 𝐴 βˆ€π‘Ÿ ∈ 𝑠 (π‘Ÿπ‘…π‘Ž ∨ π‘Ÿ = π‘Ž))) β†’ βˆƒπ‘Ž ∈ 𝐴 βˆ€π‘ ∈ 𝐴 Β¬ π‘Žπ‘…π‘)
 
Theoremzorn2g 10373* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10376 avoids the Axiom of Choice by assuming that 𝐴 is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ βˆ€π‘€((𝑀 βŠ† 𝐴 ∧ 𝑅 Or 𝑀) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘§ ∈ 𝑀 (𝑧𝑅π‘₯ ∨ 𝑧 = π‘₯))) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯𝑅𝑦)
 
Theoremzorng 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 𝐴 is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
((𝐴 ∈ dom card ∧ βˆ€π‘§((𝑧 βŠ† 𝐴 ∧ [⊊] Or 𝑧) β†’ βˆͺ 𝑧 ∈ 𝐴)) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ ⊊ 𝑦)
 
Theoremzornn0g 10375* Variant of Zorn's lemma zorng 10374 in which βˆ…, the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)
((𝐴 ∈ dom card ∧ 𝐴 β‰  βˆ… ∧ βˆ€π‘§((𝑧 βŠ† 𝐴 ∧ 𝑧 β‰  βˆ… ∧ [⊊] Or 𝑧) β†’ βˆͺ 𝑧 ∈ 𝐴)) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ ⊊ 𝑦)
 
Theoremzorn2 10376* Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set 𝐴 (with an ordering relation 𝑅) 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.)
𝐴 ∈ V    β‡’   ((𝑅 Po 𝐴 ∧ βˆ€π‘€((𝑀 βŠ† 𝐴 ∧ 𝑅 Or 𝑀) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘§ ∈ 𝑀 (𝑧𝑅π‘₯ ∨ 𝑧 = π‘₯))) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯𝑅𝑦)
 
Theoremzorn 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.)
𝐴 ∈ V    β‡’   (βˆ€π‘§((𝑧 βŠ† 𝐴 ∧ [⊊] Or 𝑧) β†’ βˆͺ 𝑧 ∈ 𝐴) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ ⊊ 𝑦)
 
Theoremzornn0 10378* Variant of Zorn's lemma zorn 10377 in which βˆ…, the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐴 ∈ V    β‡’   ((𝐴 β‰  βˆ… ∧ βˆ€π‘§((𝑧 βŠ† 𝐴 ∧ 𝑧 β‰  βˆ… ∧ [⊊] Or 𝑧) β†’ βˆͺ 𝑧 ∈ 𝐴)) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ ⊊ 𝑦)
 
Theoremttukeylem1 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.)
(πœ‘ β†’ 𝐹:(cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))–1-1-ontoβ†’(βˆͺ 𝐴 βˆ– 𝐡))    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴))    β‡’   (πœ‘ β†’ (𝐢 ∈ 𝐴 ↔ (𝒫 𝐢 ∩ Fin) βŠ† 𝐴))
 
Theoremttukeylem2 10380* Lemma for ttukey 10388. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
(πœ‘ β†’ 𝐹:(cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))–1-1-ontoβ†’(βˆͺ 𝐴 βˆ– 𝐡))    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴))    β‡’   ((πœ‘ ∧ (𝐢 ∈ 𝐴 ∧ 𝐷 βŠ† 𝐢)) β†’ 𝐷 ∈ 𝐴)
 
Theoremttukeylem3 10381* Lemma for ttukey 10388. (Contributed by Mario Carneiro, 11-May-2015.)
(πœ‘ β†’ 𝐹:(cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))–1-1-ontoβ†’(βˆͺ 𝐴 βˆ– 𝐡))    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴))    &   πΊ = recs((𝑧 ∈ V ↦ if(dom 𝑧 = βˆͺ dom 𝑧, if(dom 𝑧 = βˆ…, 𝐡, βˆͺ ran 𝑧), ((π‘§β€˜βˆͺ dom 𝑧) βˆͺ if(((π‘§β€˜βˆͺ dom 𝑧) βˆͺ {(πΉβ€˜βˆͺ dom 𝑧)}) ∈ 𝐴, {(πΉβ€˜βˆͺ dom 𝑧)}, βˆ…)))))    β‡’   ((πœ‘ ∧ 𝐢 ∈ On) β†’ (πΊβ€˜πΆ) = if(𝐢 = βˆͺ 𝐢, if(𝐢 = βˆ…, 𝐡, βˆͺ (𝐺 β€œ 𝐢)), ((πΊβ€˜βˆͺ 𝐢) βˆͺ if(((πΊβ€˜βˆͺ 𝐢) βˆͺ {(πΉβ€˜βˆͺ 𝐢)}) ∈ 𝐴, {(πΉβ€˜βˆͺ 𝐢)}, βˆ…))))
 
Theoremttukeylem4 10382* Lemma for ttukey 10388. (Contributed by Mario Carneiro, 15-May-2015.)
(πœ‘ β†’ 𝐹:(cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))–1-1-ontoβ†’(βˆͺ 𝐴 βˆ– 𝐡))    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴))    &   πΊ = recs((𝑧 ∈ V ↦ if(dom 𝑧 = βˆͺ dom 𝑧, if(dom 𝑧 = βˆ…, 𝐡, βˆͺ ran 𝑧), ((π‘§β€˜βˆͺ dom 𝑧) βˆͺ if(((π‘§β€˜βˆͺ dom 𝑧) βˆͺ {(πΉβ€˜βˆͺ dom 𝑧)}) ∈ 𝐴, {(πΉβ€˜βˆͺ dom 𝑧)}, βˆ…)))))    β‡’   (πœ‘ β†’ (πΊβ€˜βˆ…) = 𝐡)
 
Theoremttukeylem5 10383* Lemma for ttukey 10388. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
(πœ‘ β†’ 𝐹:(cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))–1-1-ontoβ†’(βˆͺ 𝐴 βˆ– 𝐡))    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴))    &   πΊ = recs((𝑧 ∈ V ↦ if(dom 𝑧 = βˆͺ dom 𝑧, if(dom 𝑧 = βˆ…, 𝐡, βˆͺ ran 𝑧), ((π‘§β€˜βˆͺ dom 𝑧) βˆͺ if(((π‘§β€˜βˆͺ dom 𝑧) βˆͺ {(πΉβ€˜βˆͺ dom 𝑧)}) ∈ 𝐴, {(πΉβ€˜βˆͺ dom 𝑧)}, βˆ…)))))    β‡’   ((πœ‘ ∧ (𝐢 ∈ On ∧ 𝐷 ∈ On ∧ 𝐢 βŠ† 𝐷)) β†’ (πΊβ€˜πΆ) βŠ† (πΊβ€˜π·))
 
Theoremttukeylem6 10384* Lemma for ttukey 10388. (Contributed by Mario Carneiro, 15-May-2015.)
(πœ‘ β†’ 𝐹:(cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))–1-1-ontoβ†’(βˆͺ 𝐴 βˆ– 𝐡))    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴))    &   πΊ = recs((𝑧 ∈ V ↦ if(dom 𝑧 = βˆͺ dom 𝑧, if(dom 𝑧 = βˆ…, 𝐡, βˆͺ ran 𝑧), ((π‘§β€˜βˆͺ dom 𝑧) βˆͺ if(((π‘§β€˜βˆͺ dom 𝑧) βˆͺ {(πΉβ€˜βˆͺ dom 𝑧)}) ∈ 𝐴, {(πΉβ€˜βˆͺ dom 𝑧)}, βˆ…)))))    β‡’   ((πœ‘ ∧ 𝐢 ∈ suc (cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))) β†’ (πΊβ€˜πΆ) ∈ 𝐴)
 
Theoremttukeylem7 10385* Lemma for ttukey 10388. (Contributed by Mario Carneiro, 15-May-2015.)
(πœ‘ β†’ 𝐹:(cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))–1-1-ontoβ†’(βˆͺ 𝐴 βˆ– 𝐡))    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴))    &   πΊ = recs((𝑧 ∈ V ↦ if(dom 𝑧 = βˆͺ dom 𝑧, if(dom 𝑧 = βˆ…, 𝐡, βˆͺ ran 𝑧), ((π‘§β€˜βˆͺ dom 𝑧) βˆͺ if(((π‘§β€˜βˆͺ dom 𝑧) βˆͺ {(πΉβ€˜βˆͺ dom 𝑧)}) ∈ 𝐴, {(πΉβ€˜βˆͺ dom 𝑧)}, βˆ…)))))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (𝐡 βŠ† π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ ⊊ 𝑦))
 
Theoremttukey2g 10386* The TeichmΓΌller-Tukey Lemma ttukey 10388 with a slightly stronger conclusion: we can set up the maximal element of 𝐴 so that it also contains some given 𝐡 ∈ 𝐴 as a subset. (Contributed by Mario Carneiro, 15-May-2015.)
((βˆͺ 𝐴 ∈ dom card ∧ 𝐡 ∈ 𝐴 ∧ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴)) β†’ βˆƒπ‘₯ ∈ 𝐴 (𝐡 βŠ† π‘₯ ∧ βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ ⊊ 𝑦))
 
Theoremttukeyg 10387* The TeichmΓΌller-Tukey Lemma ttukey 10388 stated with the "choice" as an antecedent (the hypothesis βˆͺ 𝐴 ∈ dom card says that βˆͺ 𝐴 is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.)
((βˆͺ 𝐴 ∈ dom card ∧ 𝐴 β‰  βˆ… ∧ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴)) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ ⊊ 𝑦)
 
Theoremttukey 10388* The TeichmΓΌller-Tukey Lemma, an Axiom of Choice equivalent. If 𝐴 is a nonempty collection of finite character, then 𝐴 has a maximal element with respect to inclusion. Here "finite character" means that π‘₯ ∈ 𝐴 iff every finite subset of π‘₯ is in 𝐴. (Contributed by Mario Carneiro, 15-May-2015.)
𝐴 ∈ V    β‡’   ((𝐴 β‰  βˆ… ∧ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴)) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ ⊊ 𝑦)
 
Theoremaxdclem 10389* Lemma for axdc 10391. (Contributed by Mario Carneiro, 25-Jan-2013.)
𝐹 = (rec((𝑦 ∈ V ↦ (π‘”β€˜{𝑧 ∣ 𝑦π‘₯𝑧})), 𝑠) β†Ύ Ο‰)    β‡’   ((βˆ€π‘¦ ∈ 𝒫 dom π‘₯(𝑦 β‰  βˆ… β†’ (π‘”β€˜π‘¦) ∈ 𝑦) ∧ ran π‘₯ βŠ† dom π‘₯ ∧ βˆƒπ‘§(πΉβ€˜πΎ)π‘₯𝑧) β†’ (𝐾 ∈ Ο‰ β†’ (πΉβ€˜πΎ)π‘₯(πΉβ€˜suc 𝐾)))
 
Theoremaxdclem2 10390* Lemma for axdc 10391. Using the full Axiom of Choice, we can construct a choice function 𝑔 on 𝒫 dom π‘₯. From this, we can build a sequence 𝐹 starting at any value 𝑠 ∈ dom π‘₯ by repeatedly applying 𝑔 to the set (πΉβ€˜π‘₯) (where π‘₯ is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)
𝐹 = (rec((𝑦 ∈ V ↦ (π‘”β€˜{𝑧 ∣ 𝑦π‘₯𝑧})), 𝑠) β†Ύ Ο‰)    β‡’   (βˆƒπ‘§ 𝑠π‘₯𝑧 β†’ (ran π‘₯ βŠ† dom π‘₯ β†’ βˆƒπ‘“βˆ€π‘› ∈ Ο‰ (π‘“β€˜π‘›)π‘₯(π‘“β€˜suc 𝑛)))
 
Theoremaxdc 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.)
((βˆƒπ‘¦βˆƒπ‘§ 𝑦π‘₯𝑧 ∧ ran π‘₯ βŠ† dom π‘₯) β†’ βˆƒπ‘“βˆ€π‘› ∈ Ο‰ (π‘“β€˜π‘›)π‘₯(π‘“β€˜suc 𝑛))
 
Theoremfodomg 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.)
(𝐴 ∈ 𝑉 β†’ (𝐹:𝐴–onto→𝐡 β†’ 𝐡 β‰Ό 𝐴))
 
Theoremfodom 10393 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
𝐴 ∈ V    β‡’   (𝐹:𝐴–onto→𝐡 β†’ 𝐡 β‰Ό 𝐴)
 
Theoremdmct 10394 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 β‰Ό Ο‰ β†’ dom 𝐴 β‰Ό Ο‰)
 
Theoremrnct 10395 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 β‰Ό Ο‰ β†’ ran 𝐴 β‰Ό Ο‰)
 
Theoremfodomb 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.)
((𝐴 β‰  βˆ… ∧ βˆƒπ‘“ 𝑓:𝐴–onto→𝐡) ↔ (βˆ… β‰Ί 𝐡 ∧ 𝐡 β‰Ό 𝐴))
 
Theoremwdomac 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.)
(𝑋 β‰Ό* π‘Œ ↔ 𝑋 β‰Ό π‘Œ)
 
Theorembrdom3 10398* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)
𝐡 ∈ V    β‡’   (𝐴 β‰Ό 𝐡 ↔ βˆƒπ‘“(βˆ€π‘₯βˆƒ*𝑦 π‘₯𝑓𝑦 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 𝑦𝑓π‘₯))
 
Theorembrdom5 10399* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)
𝐡 ∈ V    β‡’   (𝐴 β‰Ό 𝐡 ↔ βˆƒπ‘“(βˆ€π‘₯ ∈ 𝐡 βˆƒ*𝑦 π‘₯𝑓𝑦 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 𝑦𝑓π‘₯))
 
Theorembrdom4 10400* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
𝐡 ∈ V    β‡’   (𝐴 β‰Ό 𝐡 ↔ βˆƒπ‘“(βˆ€π‘₯ ∈ 𝐡 βˆƒ*𝑦 ∈ 𝐴 π‘₯𝑓𝑦 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 𝑦𝑓π‘₯))
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