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Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfin34 9801 Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝐴 ∈ FinIII𝐴 ∈ FinIV)

Theoremisfin5-2 9802 Alternate definition of V-finite which emphasizes the idempotent behavior of V-infinite sets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴𝐴))))

Theoremfin45 9803 Every IV-finite set is V-finite: if we can pack two copies of the set into itself, we can certainly leave space. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 18-May-2015.)
(𝐴 ∈ FinIV𝐴 ∈ FinV)

Theoremfin56 9804 Every V-finite set is VI-finite because multiplication dominates addition for cardinals. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ FinV𝐴 ∈ FinVI)

Theoremfin17 9805 Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ FinVII)

Theoremfin67 9806 Every VI-finite set is VII-finite. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ FinVI𝐴 ∈ FinVII)

Theoremisfin7-2 9807 A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))

Theoremfin71num 9808 A well-orderable set is VII-finite iff it is I-finite. Thus, even without choice, on the class of well-orderable sets all eight definitions of finite set coincide. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴 ∈ dom card → (𝐴 ∈ FinVII𝐴 ∈ Fin))

Theoremdffin7-2 9809 Class form of isfin7-2 9807. (Contributed by Mario Carneiro, 17-May-2015.)
FinVII = (Fin ∪ (V ∖ dom card))

Theoremdfacfin7 9810 Axiom of Choice equivalent: the VII-finite sets are the same as I-finite sets. (Contributed by Mario Carneiro, 18-May-2015.)
(CHOICE ↔ FinVII = Fin)

Theoremfin1a2lem1 9811 Lemma for fin1a2 9826. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       (𝐴 ∈ On → (𝑆𝐴) = suc 𝐴)

Theoremfin1a2lem2 9812 Lemma for fin1a2 9826. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       𝑆:On–1-1→On

Theoremfin1a2lem3 9813 Lemma for fin1a2 9826. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))       (𝐴 ∈ ω → (𝐸𝐴) = (2o ·o 𝐴))

Theoremfin1a2lem4 9814 Lemma for fin1a2 9826. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))       𝐸:ω–1-1→ω

Theoremfin1a2lem5 9815 Lemma for fin1a2 9826. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))       (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))

Theoremfin1a2lem6 9816 Lemma for fin1a2 9826. Establish that ω can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))    &   𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)

Theoremfin1a2lem7 9817* Lemma for fin1a2 9826. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))    &   𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       ((𝐴𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)

Theoremfin1a2lem8 9818* Lemma for fin1a2 9826. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
((𝐴𝑉 ∧ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ FinIII ∨ (𝐴𝑥) ∈ FinIII)) → 𝐴 ∈ FinIII)

Theoremfin1a2lem9 9819* Lemma for fin1a2 9826. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)
(( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → {𝑏𝑋𝑏𝐴} ∈ Fin)

Theoremfin1a2lem10 9820 Lemma for fin1a2 9826. A nonempty finite union of members of a chain is a member of the chain. (Contributed by Stefan O'Rear, 8-Nov-2014.)
((𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ∧ [] Or 𝐴) → 𝐴𝐴)

Theoremfin1a2lem11 9821* Lemma for fin1a2 9826. (Contributed by Stefan O'Rear, 8-Nov-2014.)
(( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝐴 ∪ {∅}))

Theoremfin1a2lem12 9822 Lemma for fin1a2 9826. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ¬ 𝐵 ∈ FinIII)

Theoremfin1a2lem13 9823 Lemma for fin1a2 9826. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) → ¬ (𝐵𝐶) ∈ FinII)

Theoremfin12 9824 Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 9826. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ FinII)

Theoremfin1a2s 9825* 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.)
((𝐴𝑉 ∧ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ FinII)) → 𝐴 ∈ FinII)

Theoremfin1a2 9826 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.)
(𝐴 ∈ FinIa𝐴 ∈ FinII)

2.6.14  Hereditarily size-limited sets without Choice

Theoremitunifval 9827* 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 9828* 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 9829* 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 9830* 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 9831* 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 9832* The union of all union iterates creates the transitive closure; compare trcl 9158. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (TC‘𝐴) = ran (𝑈𝐴)

Theoremituniiun 9833* 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 9834* Lemma for hsmex 9843. 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 9835* Lemma for hsmex 9843. 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 9836* Lemma for hsmex 9843. 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 9837 Lemma for hsmex 9843. 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 9838* Lemma for hsmex 9843. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9986 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 9839* Lemma for hsmex 9843. 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 9840* Lemma for hsmex 9843. 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 9841* Lemma for hsmex 9843. Combining the above constraints, along with itunitc 9832 and tcrank 9301, 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 9842* Lemma for hsmex 9843. (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 9843* 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 9044. (Contributed by Stefan O'Rear, 14-Feb-2015.)
(𝑋𝑉 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)

Theoremhsmex2 9844* The set of hereditary size-limited sets, assuming ax-reg 9044. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)

Theoremhsmex3 9845* The set of hereditary size-limited sets, assuming ax-reg 9044, 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 9870, as well as weaker forms such as the axiom of countable choice ax-cc 9846 and dependent choice ax-dc 9857. 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 9846* The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 9888, but is weak enough that it can be proven using DC (see axcc 9869). 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 9847* Lemma for axcc2 9848. (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 9848* 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 9849* A possibly more useful version of ax-cc 9846 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 9850* A version of axcc3 9849 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 9851 An ax-cc 9846 equivalent: every set has choice sets of length ω. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC ω = V

Theoremaxcc4dom 9852* Relax the constraint on axcc4 9850 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 9853* Lemma for domtriom 9854. (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 9854 Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 9725) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V       (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)

Theoremfin41 9855 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 9856 A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 9846. See dominfac 9984 for a version proved from ax-ac 9870. The axiom of Regularity is used for this proof, via inf3lem6 9084, 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 9857* 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 9932. 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 9858 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 9859* Lemma for axdc2 9860. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹𝑥), and show that the "function" described by ax-dc 9857 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 9860* An apparent strengthening of ax-dc 9857 (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 9861* 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).) (Contributed by Mario Carneiro, 27-Jan-2013.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 18-Mar-2014.)
𝐴 ∈ V    &   𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}       𝑆 ∈ V

Theoremaxdc3lem2 9862* Lemma for axdc3 9865. 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 9857 (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 9857 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 9863* Simple substitution lemma for axdc3 9865. (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 9864* Lemma for axdc3 9865. 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 9857 (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 9857 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 9860. See axdc3lem2 9862 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 9865* 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 9866* Lemma for axdc4 9867. (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 9867* A more general version of axdc3 9865 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 9868* Lemma for axcc 9869. (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 9869* Although CC can be proven trivially using ac5 9888, 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 9870* 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 9873 for a more detailed explanation. Theorem ac2 9872 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 9876 is slightly shorter when the biconditional of ax-ac 9870 is expanded into implication and negation. In axac3 9875 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10092 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 9903, ac5 9888, and ac7 9884. The Axiom of Regularity ax-reg 9044 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2b 9545. Equivalents to AC are the well-ordering theorem weth 9906 and Zorn's lemma zorn 9918. See ac4 9886 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 9044 for derivation of AC equivalents, we provide ax-ac2 9874 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 9874 from ax-ac 9870 is shown by theorem axac2 9877, and the reverse derivation by axac 9878. Therefore, new proofs should normally use ax-ac2 9874 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

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

Theoremzfac 9871* Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 9870. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
𝑥𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Theoremac2 9872* 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 9873 is easier to understand.) Note: aceq0 9533 shows the logical equivalence to ax-ac 9870. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)

Theoremac3 9873* Axiom of Choice using abbreviations. The logical equivalence to ax-ac 9870 can be established by chaining aceq0 9533 and aceq2 9534. A standard textbook version of AC is derived from this one in dfac2a 9544, and this version of AC is derived from the textbook version in dfac2b 9545, showing their logical equivalence (see dfac2 9546).

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 9069. The key theorem for this (used in the proof of dfac2b 9545) is preleq 9067. 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 9874* In order to avoid uses of ax-reg 9044 for derivation of AC equivalents, we provide ax-ac2 9874, 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 9876. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1797 available. The derivation of ax-ac2 9874 from ax-ac 9870 is shown by theorem axac2 9877, and the reverse derivation by axac 9878. Note that we use ax-reg 9044 to derive ax-ac 9870 from ax-ac2 9874, but not to derive ax-ac2 9874 from ax-ac 9870. (Contributed by NM, 19-Dec-2016.)
𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))

Theoremaxac3 9875 This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 9874 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
CHOICE

Theoremackm 9876* 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 9581. 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 9581).

The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html 9581 http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html 9581. (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 9877* Derive ax-ac2 9874 from ax-ac 9870. (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 9878* Derive ax-ac 9870 from ax-ac2 9874. Note that ax-reg 9044 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)
𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))

Theoremaxaci 9879 Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥𝜑)       𝜑

Theoremcardeqv 9880 All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
dom card = V

Theoremnumth3 9881 All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
(𝐴𝑉𝐴 ∈ dom card)

Theoremnumth2 9882* 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 9883* 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 9884* 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 9885* 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 9886* Equivalent of Axiom of Choice. We do not insist that 𝑓 be a function. However, theorem ac5 9888, 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 9902. (Contributed by NM, 21-Jul-1996.)

𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)

Theoremac4c 9887* Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.)
𝐴 ∈ V       𝑓𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)

Theoremac5 9888* 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 9886. (Contributed by NM, 29-Aug-1999.)
𝐴 ∈ V       𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))

Theoremac5b 9889* Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)
𝐴 ∈ V       (∀𝑥𝐴 𝑥 ≠ ∅ → ∃𝑓(𝑓:𝐴 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))

Theoremac6num 9890* A version of ac6 9891 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 9891* 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 9895, 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 9892* 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 9893* 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 9894* 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 9895* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9310, we derive this strong version of ac6 9891 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 9896* Equivalent of Axiom of Choice. Contrapositive of ac6s 9895. (Contributed by NM, 10-Jun-2007.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑓(𝑓:𝐴𝐵 → ∃𝑥𝐴 𝜓) → ∃𝑥𝐴𝑦𝐵 𝜑)

Theoremac6s2 9897* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 9898. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 𝜓))

Theoremac6s3 9898* 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 9899* ac6s 9895 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))

Theoremac6sf 9900* Version of ac6 9891 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)
𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

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