P. 105 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10401-10500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fin1a2lem3 10401	Lemma for fin1a2 10414. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))    ⇒   ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴))
Theorem	fin1a2lem4 10402	Lemma for fin1a2 10414. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))    ⇒   ⊢ 𝐸:ω–1-1→ω
Theorem	fin1a2lem5 10403	Lemma for fin1a2 10414. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))    ⇒   ⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))
Theorem	fin1a2lem6 10404	Lemma for fin1a2 10414. 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 𝐸–1-1-onto→(ω ∖ ran 𝐸)
Theorem	fin1a2lem7 10405*	Lemma for fin1a2 10414. 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)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 ∖ 𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)
Theorem	fin1a2lem8 10406*	Lemma for fin1a2 10414. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ FinIII ∨ (𝐴 ∖ 𝑥) ∈ FinIII)) → 𝐴 ∈ FinIII)
Theorem	fin1a2lem9 10407*	Lemma for fin1a2 10414. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)
⊢ (( [⊊] Or 𝑋 ∧ 𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → {𝑏 ∈ 𝑋 ∣ 𝑏 ≼ 𝐴} ∈ Fin)
Theorem	fin1a2lem10 10408	Lemma for fin1a2 10414. 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) → ∪ 𝐴 ∈ 𝐴)
Theorem	fin1a2lem11 10409*	Lemma for fin1a2 10414. (Contributed by Stefan O'Rear, 8-Nov-2014.)
⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝐴 ∪ {∅}))
Theorem	fin1a2lem12 10410	Lemma for fin1a2 10414. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ¬ 𝐵 ∈ FinIII)
Theorem	fin1a2lem13 10411	Lemma for fin1a2 10414. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 ∖ 𝐶) ∈ FinII)
Theorem	fin12 10412	Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 10414. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinII)
Theorem	fin1a2s 10413*	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ FinII)) → 𝐴 ∈ FinII)
Theorem	fin1a2 10414	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 ∈ FinIa → 𝐴 ∈ FinII)
2.6.17  Hereditarily size-limited sets without Choice
Theorem	itunifval 10415*	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))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))
Theorem	itunifn 10416*	Functionality of the iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝑈‘𝐴) Fn ω)
Theorem	ituni0 10417*	A zero-fold iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ((𝑈‘𝐴)‘∅) = 𝐴)
Theorem	itunisuc 10418*	Successor iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))    ⇒   ⊢ ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵)
Theorem	itunitc1 10419*	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))    ⇒   ⊢ ((𝑈‘𝐴)‘𝐵) ⊆ (TC‘𝐴)
Theorem	itunitc 10420*	The union of all union iterates creates the transitive closure; compare trcl 9727. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))    ⇒   ⊢ (TC‘𝐴) = ∪ ran (𝑈‘𝐴)
Theorem	ituniiun 10421*	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))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ((𝑈‘𝐴)‘suc 𝐵) = ∪ 𝑎 ∈ 𝐴 ((𝑈‘𝑎)‘𝐵))
Theorem	hsmexlem7 10422*	Lemma for hsmex 10431. 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)
Theorem	hsmexlem8 10423*	Lemma for hsmex 10431. 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 × (𝐻‘𝑎))))
Theorem	hsmexlem9 10424*	Lemma for hsmex 10431. 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‘𝑎) ∈ On)
Theorem	hsmexlem1 10425	Lemma for hsmex 10431. 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))
Theorem	hsmexlem2 10426*	Lemma for hsmex 10431. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 10574 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 , ∪ 𝑎 ∈ 𝐴 𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
Theorem	hsmexlem3 10427*	Lemma for hsmex 10431. 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 , ∪ 𝑎 ∈ 𝐴 𝐵)    ⇒   ⊢ (((𝐴 ≼* 𝐷 ∧ 𝐶 ∈ On) ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐷 × 𝐶)))
Theorem	hsmexlem4 10428*	Lemma for hsmex 10431. 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})𝑏 ≼ 𝑋}    &   ⊢ 𝑂 = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐)))    ⇒   ⊢ ((𝑐 ∈ ω ∧ 𝑑 ∈ 𝑆) → dom 𝑂 ∈ (𝐻‘𝑐))
Theorem	hsmexlem5 10429*	Lemma for hsmex 10431. Combining the above constraints, along with itunitc 10420 and tcrank 9883, 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})𝑏 ≼ 𝑋}    &   ⊢ 𝑂 = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐)))    ⇒   ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))
Theorem	hsmexlem6 10430*	Lemma for hsmex 10431. (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})𝑏 ≼ 𝑋}    &   ⊢ 𝑂 = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐)))    ⇒   ⊢ 𝑆 ∈ V
Theorem	hsmex 10431*	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 9591. (Contributed by Stefan O'Rear, 14-Feb-2015.)
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V)
Theorem	hsmex2 10432*	The set of hereditary size-limited sets, assuming ax-reg 9591. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V)
Theorem	hsmex3 10433*	The set of hereditary size-limited sets, assuming ax-reg 9591, using strict comparison (an easy corollary by separation). (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≺ 𝑋} ∈ V)
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY In this section we add the Axiom of Choice ax-ac 10458, as well as weaker forms such as the axiom of countable choice ax-cc 10434 and dependent choice ax-dc 10445. 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
Axiom	ax-cc 10434*	The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10476, but is weak enough that it can be proven using DC (see axcc 10457). 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 → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
Theorem	axcc2lem 10435*	Lemma for axcc2 10436. (Contributed by Mario Carneiro, 8-Feb-2013.)
⊢ 𝐾 = (𝑛 ∈ ω ↦ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))    &   ⊢ 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴‘𝑛))))    ⇒   ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
Theorem	axcc2 10436*	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 ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
Theorem	axcc3 10437*	A possibly more useful version of ax-cc 10434 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 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))
Theorem	axcc4 10438*	A version of axcc3 10437 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝑁 ≈ ω    &   ⊢ (𝑥 = (𝑓‘𝑛) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓))
Theorem	acncc 10439	An ax-cc 10434 equivalent: every set has choice sets of length ω. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ AC ω = V
Theorem	axcc4dom 10440*	Relax the constraint on axcc4 10438 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = (𝑓‘𝑛) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝑁 ≼ ω ∧ ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑) → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓))
Theorem	domtriomlem 10441*	Lemma for domtriom 10442. (Contributed by Mario Carneiro, 9-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 = {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)}    &   ⊢ 𝐶 = (𝑛 ∈ ω ↦ ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)))    ⇒   ⊢ (¬ 𝐴 ∈ Fin → ω ≼ 𝐴)
Theorem	domtriom 10442	Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 10313) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)
Theorem	fin41 10443	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
Theorem	dominf 10444	A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 10434. See dominfac 10572 for a version proved from ax-ac 10458. The axiom of Regularity is used for this proof, via inf3lem6 9632, 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
Axiom	ax-dc 10445*	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 10520. 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.)
⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
Theorem	dcomex 10446	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
Theorem	axdc2lem 10447*	Lemma for axdc2 10448. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹‘𝑥), and show that the "function" described by ax-dc 10445 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⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}    &   ⊢ 𝐺 = (𝑥 ∈ ω ↦ (ℎ‘𝑥))    ⇒   ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))
Theorem	axdc2 10448*	An apparent strengthening of ax-dc 10445 (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‘(𝑔‘𝑘))))
Theorem	axdc3lem 10449*	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 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))}    ⇒   ⊢ 𝑆 ∈ V
Theorem	axdc3lem2 10450*	Lemma for axdc3 10453. 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 10445 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (ℎ‘𝑛):𝑚⟶𝐴 (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 10445 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 ℎ, 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 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))}    &   ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})    ⇒   ⊢ (∃ℎ(ℎ:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (ℎ‘suc 𝑘) ∈ (𝐺‘(ℎ‘𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))
Theorem	axdc3lem3 10451*	Simple substitution lemma for axdc3 10453. (Contributed by Mario Carneiro, 27-Jan-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))}    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚⟶𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵‘𝑘))))
Theorem	axdc3lem4 10452*	Lemma for axdc3 10453. 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 10445 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (ℎ‘𝑛):𝑚⟶𝐴 (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 10445 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 10448. See axdc3lem2 10450 for the rest of the proof. (Contributed by Mario Carneiro, 27-Jan-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))}    &   ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))
Theorem	axdc3 10453*	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    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))
Theorem	axdc4lem 10454*	Lemma for axdc4 10455. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐺 = (𝑛 ∈ ω, 𝑥 ∈ 𝐴 ↦ ({suc 𝑛} × (𝑛𝐹𝑥)))    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝑘𝐹(𝑔‘𝑘))))
Theorem	axdc4 10455*	A more general version of axdc3 10453 that allows the function 𝐹 to vary with 𝑘. (Contributed by Mario Carneiro, 31-Jan-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝑘𝐹(𝑔‘𝑘))))
Theorem	axcclem 10456*	Lemma for axcc 10457. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
⊢ 𝐴 = (𝑥 ∖ {∅})    &   ⊢ 𝐹 = (𝑛 ∈ ω, 𝑦 ∈ ∪ 𝐴 ↦ (𝑓‘𝑛))    &   ⊢ 𝐺 = (𝑤 ∈ 𝐴 ↦ (ℎ‘suc (◡𝑓‘𝑤)))    ⇒   ⊢ (𝑥 ≈ ω → ∃𝑔∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑔‘𝑧) ∈ 𝑧))
Theorem	axcc 10457*	Although CC can be proven trivially using ac5 10476, we prove it here using DC. (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Feb-2013.)
⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
3.2  ZFC Set Theory - add the Axiom of Choice
3.2.1  Introduce the Axiom of Choice
Axiom	ax-ac 10458*	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 10461 for a more detailed explanation. Theorem ac2 10460 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 10464 is slightly shorter when the biconditional of ax-ac 10458 is expanded into implication and negation. In axac3 10463 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10680 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 10491, ac5 10476, and ac7 10472. The Axiom of Regularity ax-reg 9591 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as Theorem dfac2b 10129. Equivalents to AC are the well-ordering theorem weth 10494 and Zorn's lemma zorn 10506. See ac4 10474 for comments about stronger versions of AC. In order to avoid uses of ax-reg 9591 for derivation of AC equivalents, we provide ax-ac2 10462 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 10462 from ax-ac 10458 is shown by Theorem axac2 10465, and the reverse derivation by axac 10466. Therefore, new proofs should normally use ax-ac2 10462 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
Theorem	zfac 10459*	Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 10458. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
⊢ ∃𝑥∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))
Theorem	ac2 10460*	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 10461 is easier to understand.) Note: aceq0 10117 shows the logical equivalence to ax-ac 10458. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
⊢ ∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)
Theorem	ac3 10461*	Axiom of Choice using abbreviations. The logical equivalence to ax-ac 10458 can be established by chaining aceq0 10117 and aceq2 10118. A standard textbook version of AC is derived from this one in dfac2a 10128, and this version of AC is derived from the textbook version in dfac2b 10129, showing their logical equivalence (see dfac2 10130). 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 9617. The key theorem for this (used in the proof of dfac2b 10129) is preleq 9615. 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.)
⊢ ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
Axiom	ax-ac2 10462*	In order to avoid uses of ax-reg 9591 for derivation of AC equivalents, we provide ax-ac2 10462, 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 10464. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1796 available. The derivation of ax-ac2 10462 from ax-ac 10458 is shown by Theorem axac2 10465, and the reverse derivation by axac 10466. Note that we use ax-reg 9591 to derive ax-ac 10458 from ax-ac2 10462, but not to derive ax-ac2 10462 from ax-ac 10458. (Contributed by NM, 19-Dec-2016.)
⊢ ∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))
Theorem	axac3 10463	This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10462 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
⊢ CHOICE
Theorem	ackm 10464*	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 10165. 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 10165. The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html 10165 http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html 10165. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)
⊢ ∀𝑥∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))
Theorem	axac2 10465*	Derive ax-ac2 10462 from ax-ac 10458. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))
Theorem	axac 10466*	Derive ax-ac 10458 from ax-ac2 10462. Note that ax-reg 9591 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)
⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
Theorem	axaci 10467	Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ (CHOICE ↔ ∀𝑥𝜑)    ⇒   ⊢ 𝜑
Theorem	cardeqv 10468	All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ dom card = V
Theorem	numth3 10469	All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card)
Theorem	numth2 10470*	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 𝑥 ≈ 𝐴
Theorem	numth 10471*	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:𝑥–1-1-onto→𝐴
Theorem	ac7 10472*	An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.)
⊢ ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
Theorem	ac7g 10473*	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 → ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅))
Theorem	ac4 10474*	Equivalent of Axiom of Choice. We do not insist that 𝑓 be a function. However, Theorem ac5 10476, 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" ∃𝐹∀𝑧(𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 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 10490. (Contributed by NM, 21-Jul-1996.)
⊢ ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
Theorem	ac4c 10475*	Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)
Theorem	ac5 10476*	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 10474. (Contributed by NM, 29-Aug-1999.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))
Theorem	ac5b 10477*	Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
Theorem	ac6num 10478*	A version of ac6 10479 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ac6 10479*	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 10483, 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 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ac6c4 10480*	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    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
Theorem	ac6c5 10481*	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    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)
Theorem	ac9 10482*	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    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ X𝑥 ∈ 𝐴 𝐵 ≠ ∅)
Theorem	ac6s 10483*	Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9892, we derive this strong version of ac6 10479 that doesn't require 𝐵 to be a set. (Contributed by NM, 4-Feb-2004.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ac6n 10484*	Equivalent of Axiom of Choice. Contrapositive of ac6s 10483. (Contributed by NM, 10-Jun-2007.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑓(𝑓:𝐴⟶𝐵 → ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
Theorem	ac6s2 10485*	Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 10486. (Contributed by NM, 29-Sep-2006.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ac6s3 10486*	Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 𝜓)
Theorem	ac6sg 10487*	ac6s 10483 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)
⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	ac6sf 10488*	Version of ac6 10479 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)
⊢ Ⅎ𝑦𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ac6s4 10489*	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    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
Theorem	ac6s5 10490*	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    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)
Theorem	ac8 10491*	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.)
⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
Theorem	ac9s 10492*	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 9890). (Contributed by NM, 29-Sep-2006.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ X𝑥 ∈ 𝐴 𝐵 ≠ ∅)
3.2.2  AC equivalents: well-ordering, Zorn's lemma
Theorem	numthcor 10493*	Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
Theorem	weth 10494*	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 → ∃𝑥 𝑥 We 𝐴)
Theorem	zorn2lem1 10495*	Lemma for zorn2 10505. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))    &   ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}    ⇒   ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)
Theorem	zorn2lem2 10496*	Lemma for zorn2 10505. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))    &   ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}    ⇒   ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝑦 ∈ 𝑥 → (𝐹‘𝑦)𝑅(𝐹‘𝑥)))
Theorem	zorn2lem3 10497*	Lemma for zorn2 10505. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))    &   ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}    ⇒   ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
Theorem	zorn2lem4 10498*	Lemma for zorn2 10505. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))    &   ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}    ⇒   ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)
Theorem	zorn2lem5 10499*	Lemma for zorn2 10505. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))    &   ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}    &   ⊢ 𝐻 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧}    ⇒   ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (𝐹 “ 𝑥) ⊆ 𝐴)
Theorem	zorn2lem6 10500*	Lemma for zorn2 10505. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))    &   ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}    &   ⊢ 𝐻 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧}    ⇒   ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹 “ 𝑥)))