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Theorem List for Metamath Proof Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement

2.6.13  Hereditarily size-limited sets without Choice

Theoremitunifval 9001* 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.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (𝐴𝑉 → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))

Theoremitunifn 9002* Functionality of the iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (𝐴𝑉 → (𝑈𝐴) Fn ω)

Theoremituni0 9003* A zero-fold iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (𝐴𝑉 → ((𝑈𝐴)‘∅) = 𝐴)

Theoremitunisuc 9004* Successor iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵)

Theoremitunitc1 9005* Each union iterate is a member of the transitive closure. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)

Theoremitunitc 9006* The union of all union iterates creates the transitive closure; compare trcl 8367. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (TC‘𝐴) = ran (𝑈𝐴)

Theoremituniiun 9007* Unwrap an iterated union from the "other end". (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (𝐴𝑉 → ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))

Theoremhsmexlem7 9008* Lemma for hsmex 9017. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)       (𝐻‘∅) = (har‘𝒫 𝑋)

Theoremhsmexlem8 9009* Lemma for hsmex 9017. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)       (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻𝑎))))

Theoremhsmexlem9 9010* Lemma for hsmex 9017. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)       (𝑎 ∈ ω → (𝐻𝑎) ∈ On)

Theoremhsmexlem1 9011 Lemma for hsmex 9017. 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.)
𝑂 = OrdIso( E , 𝐴)       ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))

Theoremhsmexlem2 9012* Lemma for hsmex 9017. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9156 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.)
𝐹 = OrdIso( E , 𝐵)    &   𝐺 = OrdIso( E , 𝑎𝐴 𝐵)       ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))

Theoremhsmexlem3 9013* Lemma for hsmex 9017. Clear 𝐼 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.)
𝐹 = OrdIso( E , 𝐵)    &   𝐺 = OrdIso( E , 𝑎𝐴 𝐵)       (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐷 × 𝐶)))

Theoremhsmexlem4 9014* Lemma for hsmex 9017. The core induction, establishing bounds on the order types of iterated unions of the initial set. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝑋 ∈ V    &   𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)    &   𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))    &   𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}    &   𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))       ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom 𝑂 ∈ (𝐻𝑐))

Theoremhsmexlem5 9015* Lemma for hsmex 9017. Combining the above constraints, along with itunitc 9006 and tcrank 8510, gives an effective constraint on the rank of 𝑆. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝑋 ∈ V    &   𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)    &   𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))    &   𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}    &   𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))       (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))

Theoremhsmexlem6 9016* Lemma for hsmex 9017. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝑋 ∈ V    &   𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)    &   𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))    &   𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}    &   𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))       𝑆 ∈ V

Theoremhsmex 9017* 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 8260. (Contributed by Stefan O'Rear, 14-Feb-2015.)
(𝑋𝑉 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)

Theoremhsmex2 9018* The set of hereditary size-limited sets, assuming ax-reg 8260. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)

Theoremhsmex3 9019* The set of hereditary size-limited sets, assuming ax-reg 8260, using strict comparison (an easy corollary by separation). (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)

PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY

In this section we add the Axiom of Choice ax-ac 9044, as well as weaker forms such as the axiom of countable choice ax-cc 9020 and dependent choice ax-dc 9031. 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 9020* The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 9062, but is weak enough that it can be proven using DC (see axcc 9043). 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.)
(𝑥 ≈ ω → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))

Theoremaxcc2lem 9021* Lemma for axcc2 9022. (Contributed by Mario Carneiro, 8-Feb-2013.)
𝐾 = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))    &   𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))    &   𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴𝑛))))       𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))

Theoremaxcc2 9022* 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.)
𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))

Theoremaxcc3 9023* A possibly more useful version of ax-cc 9020 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    &   𝑁 ≈ ω       𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))

Theoremaxcc4 9024* A version of axcc3 9023 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.)
𝐴 ∈ V    &   𝑁 ≈ ω    &   (𝑥 = (𝑓𝑛) → (𝜑𝜓))       (∀𝑛𝑁𝑥𝐴 𝜑 → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜓))

Theoremacncc 9025 An ax-cc 9020 equivalent: every set has choice sets of length ω. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC ω = V

Theoremaxcc4dom 9026* Relax the constraint on axcc4 9024 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.)
𝐴 ∈ V    &   (𝑥 = (𝑓𝑛) → (𝜑𝜓))       ((𝑁 ≼ ω ∧ ∀𝑛𝑁𝑥𝐴 𝜑) → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜓))

Theoremdomtriomlem 9027* Lemma for domtriom 9028. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V    &   𝐵 = {𝑦 ∣ (𝑦𝐴𝑦 ≈ 𝒫 𝑛)}    &   𝐶 = (𝑛 ∈ ω ↦ ((𝑏𝑛) ∖ 𝑘𝑛 (𝑏𝑘)))       𝐴 ∈ Fin → ω ≼ 𝐴)

Theoremdomtriom 9028 Trichotomy of equinumerosity for ω, proven using CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 8899) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V       (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)

Theoremfin41 9029 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 9030 A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 9020. See dominfac 9154 for a version proved from ax-ac 9044. The axiom of Regularity is used for this proof, via inf3lem6 8293, and its use is necessary: otherwise the set 𝐴 = {𝐴} or 𝐴 = {∅, 𝐴} (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ 𝐴 𝐴) → ω ≼ 𝐴)

3.1.2  Introduce the Axiom of Dependent Choice

Axiomax-dc 9031* 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 9106. 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.)
((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))

Theoremdcomex 9032 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.)
ω ∈ V

Theoremaxdc2lem 9033* Lemma for axdc2 9034. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹𝑥), and show that the "function" described by ax-dc 9031 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    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}    &   𝐺 = (𝑥 ∈ ω ↦ (𝑥))       ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc2 9034* An apparent strengthening of ax-dc 9031 (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(𝑥 + 1) ∈ 𝐹(𝑔(𝑥)) for all 𝑥 ∈ ω. 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       ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc3lem 9035* 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 (ω × 𝐴).) (Unnecessary distinct variable restrictions were removed by David Abernethy, 18-Mar-2014.) (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario Carneiro, 18-Mar-2014.)
𝐴 ∈ V    &   𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}       𝑆 ∈ V

Theoremaxdc3lem2 9036* Lemma for axdc3 9039. We have constructed a "candidate set" 𝑆, which consists of all finite sequences 𝑠 that satisfy our property of interest, namely 𝑠(𝑥 + 1) ∈ 𝐹(𝑠(𝑥)) on its domain, but with the added constraint that 𝑠(0) = 𝐶. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 9031 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (𝑛):𝑚𝐴 (for some integer 𝑚). 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 9031 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence , we can construct the sequence 𝑔 that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.)
𝐴 ∈ V    &   𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}    &   𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})       (∃(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc3lem3 9037* Simple substitution lemma for axdc3 9039. (Contributed by Mario Carneiro, 27-Jan-2013.)
𝐴 ∈ V    &   𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}    &   𝐵 ∈ V       (𝐵𝑆 ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))

Theoremaxdc3lem4 9038* Lemma for axdc3 9039. We have constructed a "candidate set" 𝑆, which consists of all finite sequences 𝑠 that satisfy our property of interest, namely 𝑠(𝑥 + 1) ∈ 𝐹(𝑠(𝑥)) on its domain, but with the added constraint that 𝑠(0) = 𝐶. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 9031 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (𝑛):𝑚𝐴 (for some integer 𝑚). 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 9031 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 9034. See axdc3lem2 9036 for the rest of the proof. (Contributed by Mario Carneiro, 27-Jan-2013.)
𝐴 ∈ V    &   𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}    &   𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})       ((𝐶𝐴𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc3 9039* Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value 𝐶. 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       ((𝐶𝐴𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc4lem 9040* Lemma for axdc4 9041. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
𝐴 ∈ V    &   𝐺 = (𝑛 ∈ ω, 𝑥𝐴 ↦ ({suc 𝑛} × (𝑛𝐹𝑥)))       ((𝐶𝐴𝐹:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝑘𝐹(𝑔𝑘))))

Theoremaxdc4 9041* A more general version of axdc3 9039 that allows the function 𝐹 to vary with 𝑘. (Contributed by Mario Carneiro, 31-Jan-2013.)
𝐴 ∈ V       ((𝐶𝐴𝐹:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝑘𝐹(𝑔𝑘))))

Theoremaxcclem 9042* Lemma for axcc 9043. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
𝐴 = (𝑥 ∖ {∅})    &   𝐹 = (𝑛 ∈ ω, 𝑦 𝐴 ↦ (𝑓𝑛))    &   𝐺 = (𝑤𝐴 ↦ (‘suc (𝑓𝑤)))       (𝑥 ≈ ω → ∃𝑔𝑧𝑥 (𝑧 ≠ ∅ → (𝑔𝑧) ∈ 𝑧))

Theoremaxcc 9043* Although CC can be proven trivially using ac5 9062, we prove it here using DC. (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Feb-2013.)
(𝑥 ≈ ω → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))

3.2  ZFC Set Theory - add the Axiom of Choice

3.2.1  Introduce the Axiom of Choice

Axiomax-ac 9044* 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 𝑥, there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥. One entry in the pair is the member of 𝑥, and the other entry is some arbitrary member of that member of 𝑥. See the rewritten version ac3 9047 for a more detailed explanation. Theorem ac2 9046 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 9050 is slightly shorter when the biconditional of ax-ac 9044 is expanded into implication and negation. In axac3 9049 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9262 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 9077, ac5 9062, and ac7 9058. The Axiom of Regularity ax-reg 8260 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8716. Equivalents to AC are the well-ordering theorem weth 9080 and Zorn's lemma zorn 9092. See ac4 9060 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 8260 for derivation of AC equivalents, we provide ax-ac2 9048 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 9048 from ax-ac 9044 is shown by theorem axac2 9051, and the reverse derivation by axac 9052. Therefore, new proofs should normally use ax-ac2 9048 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))

Theoremzfac 9045* Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 9044. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
𝑥𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Theoremac2 9046* 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 9047 is easier to understand.) Note: aceq0 8704 shows the logical equivalence to ax-ac 9044. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)

Theoremac3 9047* Axiom of Choice using abbreviations. The logical equivalence to ax-ac 9044 can be established by chaining aceq0 8704 and aceq2 8705. A standard textbook version of AC is derived from this one in dfac2a 8715, and this version of AC is derived from the textbook version in dfac2 8716.

The following sketch will help you understand this version of the axiom. Given any set 𝑥, the axiom says that there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥. One entry in the pair is the member of 𝑥, and the other entry is some arbitrary member of that member of 𝑥. 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 8278. The key theorem for this (used in the proof of dfac2 8716) is preleq 8277. With this modified definition of ordered pair, it can be seen that 𝑦 is actually a choice function on the members of 𝑥.

For example, suppose 𝑥 = {{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 𝑥) 𝑧 = {1, 2}, the only assignment to 𝑤 and 𝑣 that satisfies the axiom is 𝑤 = 1 and 𝑣 = {{1, 2}, 1}, so there is exactly one 𝑤 as required. We verify the other two members of 𝑥 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.)

𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣))

Axiomax-ac2 9048* In order to avoid uses of ax-reg 8260 for derivation of AC equivalents, we provide ax-ac2 9048, 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 9050. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1700 available. The derivation of ax-ac2 9048 from ax-ac 9044 is shown by theorem axac2 9051, and the reverse derivation by axac 9052. Note that we use ax-reg 8260 to derive ax-ac 9044 from ax-ac2 9048, but not to derive ax-ac2 9048 from ax-ac 9044. (Contributed by NM, 19-Dec-2016.)
𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))

Theoremaxac3 9049 This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 9048 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
CHOICE

Theoremackm 9050* A remarkable equivalent to the Axiom of Choice that has only five quantifiers (when expanded to , = primitives in prenex form), discovered and proved by Kurt Maes. This establishes a new record, reducing from 6 to 5 the largest number of quantified variables needed by any ZFC axiom. The ZF-equivalence to AC is shown by theorem dfackm 8751. 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" (http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3162v1.pdf).

The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)

𝑥𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))

Theoremaxac2 9051* Derive ax-ac2 9048 from ax-ac 9044. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))

Theoremaxac 9052* Derive ax-ac 9044 from ax-ac2 9048. Note that ax-reg 8260 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)
𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))

Theoremaxaci 9053 Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥𝜑)       𝜑

Theoremcardeqv 9054 All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
dom card = V

Theoremnumth3 9055 All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
(𝐴𝑉𝐴 ∈ dom card)

Theoremnumth2 9056* 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       𝑥 ∈ On 𝑥𝐴

Theoremnumth 9057* 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       𝑥 ∈ On ∃𝑓 𝑓:𝑥1-1-onto𝐴

Theoremac7 9058* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.)
𝑓(𝑓𝑥𝑓 Fn dom 𝑥)

Theoremac7g 9059* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)
(𝑅𝐴 → ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅))

Theoremac4 9060* Equivalent of Axiom of Choice. We do not insist that 𝑓 be a function. However, theorem ac5 9062, 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" 𝐹𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 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 9076. (Contributed by NM, 21-Jul-1996.)

𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)

Theoremac4c 9061* Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.)
𝐴 ∈ V       𝑓𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)

Theoremac5 9062* 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 9060. (Contributed by NM, 29-Aug-1999.)
𝐴 ∈ V       𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))

Theoremac5b 9063* Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)
𝐴 ∈ V       (∀𝑥𝐴 𝑥 ≠ ∅ → ∃𝑓(𝑓:𝐴 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))

Theoremac6num 9064* A version of ac6 9065 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6 9065* 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 𝐵, where 𝜑 depends on 𝑥 (the natural number) and 𝑦 (to specify a member of 𝐵). A stronger version of this theorem, ac6s 9069, allows 𝐵 to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6c4 9066* Equivalent of Axiom of Choice. 𝐵 is a collection 𝐵(𝑥) of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))

Theoremac6c5 9067* Equivalent of Axiom of Choice. 𝐵 is a collection 𝐵(𝑥) of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝐵)

Theoremac9 9068* 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    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)

Theoremac6s 9069* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 8519, we derive this strong version of ac6 9065 that doesn't require 𝐵 to be a set. (Contributed by NM, 4-Feb-2004.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6n 9070* Equivalent of Axiom of Choice. Contrapositive of ac6s 9069. (Contributed by NM, 10-Jun-2007.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑓(𝑓:𝐴𝐵 → ∃𝑥𝐴 𝜓) → ∃𝑥𝐴𝑦𝐵 𝜑)

Theoremac6s2 9071* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 9072. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 𝜓))

Theoremac6s3 9072* Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓𝑥𝐴 𝜓)

Theoremac6sg 9073* ac6s 9069 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))

Theoremac6sf 9074* Version of ac6 9065 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)
𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6s4 9075* Generalization of the Axiom of Choice to proper classes. 𝐵 is a collection 𝐵(𝑥) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))

Theoremac6s5 9076* Generalization of the Axiom of Choice to proper classes. 𝐵 is a collection 𝐵(𝑥) of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝐵)

Theoremac8 9077* An Axiom of Choice equivalent. Given a family 𝑥 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.)
((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))

Theoremac9s 9078* 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 𝐵(𝑥) (achieved via the Collection Principle cp 8517). (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)

3.2.2  AC equivalents: well-ordering, Zorn's lemma

Theoremnumthcor 9079* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)

Theoremweth 9080* 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.)
(𝐴𝑉 → ∃𝑥 𝑥 We 𝐴)

Theoremzorn2lem1 9081* Lemma for zorn2 9091. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)

Theoremzorn2lem2 9082* Lemma for zorn2 9091. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝑦𝑥 → (𝐹𝑦)𝑅(𝐹𝑥)))

Theoremzorn2lem3 9083* Lemma for zorn2 9091. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))

Theoremzorn2lem4 9084* Lemma for zorn2 9091. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑅 Po 𝐴𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)

Theoremzorn2lem5 9085* Lemma for zorn2 9091. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       (((𝑤 We 𝐴𝑥 ∈ On) ∧ ∀𝑦𝑥 𝐻 ≠ ∅) → (𝐹𝑥) ⊆ 𝐴)

Theoremzorn2lem6 9086* Lemma for zorn2 9091. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       (𝑅 Po 𝐴 → (((𝑤 We 𝐴𝑥 ∈ On) ∧ ∀𝑦𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹𝑥)))

Theoremzorn2lem7 9087* Lemma for zorn2 9091. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑠((𝑠𝐴𝑅 Or 𝑠) → ∃𝑎𝐴𝑟𝑠 (𝑟𝑅𝑎𝑟 = 𝑎))) → ∃𝑎𝐴𝑏𝐴 ¬ 𝑎𝑅𝑏)

Theoremzorn2g 9088* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 9091 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 ∧ ∀𝑤((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)

Theoremzorng 9089* 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 9092 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 ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremzornn0g 9090* Variant of Zorn's lemma zorng 9089 in which , 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 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremzorn2 9091* 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 9081 through zorn2lem7 9087; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 9087. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐴 ∈ V       ((𝑅 Po 𝐴 ∧ ∀𝑤((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)

Theoremzorn 9092* 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 9091 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
𝐴 ∈ V       (∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremzornn0 9093* Variant of Zorn's lemma zorn 9092 in which , the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremttukeylem1 9094* Lemma for ttukey 9103. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))       (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))

Theoremttukeylem2 9095* Lemma for ttukey 9103. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))       ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)

Theoremttukeylem3 9096* Lemma for ttukey 9103. (Contributed by Mario Carneiro, 11-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       ((𝜑𝐶 ∈ On) → (𝐺𝐶) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))

Theoremttukeylem4 9097* Lemma for ttukey 9103. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       (𝜑 → (𝐺‘∅) = 𝐵)

Theoremttukeylem5 9098* Lemma for ttukey 9103. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶𝐷)) → (𝐺𝐶) ⊆ (𝐺𝐷))

Theoremttukeylem6 9099* Lemma for ttukey 9103. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       ((𝜑𝐶 ∈ suc (card‘( 𝐴𝐵))) → (𝐺𝐶) ∈ 𝐴)

Theoremttukeylem7 9100* Lemma for ttukey 9103. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       (𝜑 → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))

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