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	isf32lem9 10401*	Lemma for isfin3-2 10407. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)    &   ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))    &   ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹)    &   ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)}    &   ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢))    &   ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)    &   ⊢ 𝐿 = (𝑡 ∈ 𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))))    ⇒   ⊢ (𝜑 → 𝐿:𝐺–onto→ω)
Theorem	isf32lem10 10402*	Lemma for isfin3-2 . Write in terms of weak dominance. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)    &   ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))    &   ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹)    &   ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)}    &   ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢))    &   ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)    &   ⊢ 𝐿 = (𝑡 ∈ 𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))))    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝑉 → ω ≼* 𝐺))
Theorem	isf32lem11 10403*	Lemma for isfin3-2 10407. Remove hypotheses from isf32lem10 10402. (Contributed by Stefan O'Rear, 17-May-2015.)
⊢ ((𝐺 ∈ 𝑉 ∧ (𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐺)
Theorem	isf32lem12 10404*	Lemma for isfin3-2 10407. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}    ⇒   ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → 𝐺 ∈ 𝐹))
Theorem	isfin32i 10405	One half of isfin3-2 10407. (Contributed by Mario Carneiro, 3-Jun-2015.)
⊢ (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴)
Theorem	isf33lem 10406*	Lemma for isfin3-3 10408. (Contributed by Stefan O'Rear, 17-May-2015.)
⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
Theorem	isfin3-2 10407	Weakly Dedekind-infinite sets are exactly those which can be mapped onto ω. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ¬ ω ≼* 𝐴))
Theorem	isfin3-3 10408*	Weakly Dedekind-infinite sets are exactly those with an ω-indexed descending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))
Theorem	fin33i 10409*	Inference from isfin3-3 10408. (This is actually a bit stronger than isfin3-3 10408 because it does not assume 𝐹 is a set and does not use the Axiom of Infinity either.) (Contributed by Mario Carneiro, 17-May-2015.)
⊢ ((𝐴 ∈ FinIII ∧ 𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) → ∩ ran 𝐹 ∈ ran 𝐹)
Theorem	compsscnvlem 10410*	Lemma for compsscnv 10411. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦)))
Theorem	compsscnv 10411*	Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ ◡𝐹 = 𝐹
Theorem	isf34lem1 10412*	Lemma for isfin3-4 10422. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋))
Theorem	isf34lem2 10413*	Lemma for isfin3-4 10422. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
Theorem	compssiso 10414*	Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴))
Theorem	isf34lem3 10415*	Lemma for isfin3-4 10422. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
Theorem	compss 10416*	Express image under of the complementation isomorphism. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ (𝐹 “ 𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑦) ∈ 𝐺}
Theorem	isf34lem4 10417*	Lemma for isfin3-4 10422. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ 𝑋) = ∩ (𝐹 “ 𝑋))
Theorem	isf34lem5 10418*	Lemma for isfin3-4 10422. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∩ 𝑋) = ∪ (𝐹 “ 𝑋))
Theorem	isf34lem7 10419*	Lemma for isfin3-4 10422. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ ((𝐴 ∈ FinIII ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∪ ran 𝐺 ∈ ran 𝐺)
Theorem	isf34lem6 10420*	Lemma for isfin3-4 10422. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓)))
Theorem	fin34i 10421*	Inference from isfin3-4 10422. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ ((𝐴 ∈ FinIII ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐺‘𝑥) ⊆ (𝐺‘suc 𝑥)) → ∪ ran 𝐺 ∈ ran 𝐺)
Theorem	isfin3-4 10422*	Weakly Dedekind-infinite sets are exactly those with an ω-indexed ascending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘𝑥) ⊆ (𝑓‘suc 𝑥) → ∪ ran 𝑓 ∈ ran 𝑓)))
Theorem	fin11a 10423	Every I-finite set is Ia-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinIa)
Theorem	enfin1ai 10424	Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa))
Theorem	isfin1-2 10425	A set is finite in the usual sense iff the power set of its power set is Dedekind finite. (Contributed by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ FinIV)
Theorem	isfin1-3 10426	A set is I-finite iff every system of subsets contains a maximal subset. Definition I of [Levy58] p. 2. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ ◡ [⊊] Fr 𝒫 𝐴))
Theorem	isfin1-4 10427	A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ [⊊] Fr 𝒫 𝐴))
Theorem	dffin1-5 10428	Compact quantifier-free version of the standard definition df-fin 8989. (Contributed by Stefan O'Rear, 6-Jan-2015.)
⊢ Fin = ( ≈ “ ω)
Theorem	fin23 10429	Every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets). The proof here is the only one I could find, from http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm619.pdf p.94 (writeup by Tarski, credited to Kuratowski). Translated into English and modern notation, the proof proceeds as follows (variables renamed for uniqueness): Suppose for a contradiction that 𝐴 is a set which is II-finite but not III-finite. For any countable sequence of distinct subsets 𝑇 of 𝐴, we can form a decreasing sequence of nonempty subsets (𝑈‘𝑇) by taking finite intersections of initial segments of 𝑇 while skipping over any element of 𝑇 which would cause the intersection to be empty. By II-finiteness (as fin2i2 10358) this sequence contains its intersection, call it 𝑌; since by induction every subset in the sequence 𝑈 is nonempty, the intersection must be nonempty. Suppose that an element 𝑋 of 𝑇 has nonempty intersection with 𝑌. Thus, said element has a nonempty intersection with the corresponding element of 𝑈, therefore it was used in the construction of 𝑈 and all further elements of 𝑈 are subsets of 𝑋, thus 𝑋 contains the 𝑌. That is, all elements of 𝑋 either contain 𝑌 or are disjoint from it. Since there are only two cases, there must exist an infinite subset of 𝑇 which uniformly either contain 𝑌 or are disjoint from it. In the former case we can create an infinite set by subtracting 𝑌 from each element. In either case, call the result 𝑍; this is an infinite set of subsets of 𝐴, each of which is disjoint from 𝑌 and contained in the union of 𝑇; the union of 𝑍 is strictly contained in the union of 𝑇, because only the latter is a superset of the nonempty set 𝑌. The preceding four steps may be iterated a countable number of times starting from the assumed denumerable set of subsets to produce a denumerable sequence 𝐵 of the 𝑇 sets from each stage. Great caution is required to avoid ax-dc 10486 here; in particular an effective version of the pigeonhole principle (for aleph-null pigeons and 2 holes) is required. Since a denumerable set of subsets is assumed to exist, we can conclude ω ∈ V without the axiom. This 𝐵 sequence is strictly decreasing, thus it has no minimum, contradicting the first assumption. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ FinII → 𝐴 ∈ FinIII)
Theorem	fin34 10430	Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ (𝐴 ∈ FinIII → 𝐴 ∈ FinIV)
Theorem	isfin5-2 10431	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.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴))))
Theorem	fin45 10432	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 ∈ FinIV → 𝐴 ∈ FinV)
Theorem	fin56 10433	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 ∈ FinV → 𝐴 ∈ FinVI)
Theorem	fin17 10434	Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinVII)
Theorem	fin67 10435	Every VI-finite set is VII-finite. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ FinVI → 𝐴 ∈ FinVII)
Theorem	isfin7-2 10436	A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))
Theorem	fin71num 10437	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 ∈ FinVII ↔ 𝐴 ∈ Fin))
Theorem	dffin7-2 10438	Class form of isfin7-2 10436. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ FinVII = (Fin ∪ (V ∖ dom card))
Theorem	dfacfin7 10439	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)
Theorem	fin1a2lem1 10440	Lemma for fin1a2 10455. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)    ⇒   ⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴)
Theorem	fin1a2lem2 10441	Lemma for fin1a2 10455. The successor operation on the ordinal numbers is injective or one-to-one. Lemma 1.17 of [Schloeder] p. 2. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)    ⇒   ⊢ 𝑆:On–1-1→On
Theorem	fin1a2lem3 10442	Lemma for fin1a2 10455. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))    ⇒   ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴))
Theorem	fin1a2lem4 10443	Lemma for fin1a2 10455. (Contributed by Stefan O'Rear, 7-Nov-2014.)
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))    ⇒   ⊢ 𝐸:ω–1-1→ω
Theorem	fin1a2lem5 10444	Lemma for fin1a2 10455. (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 10445	Lemma for fin1a2 10455. 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 10446*	Lemma for fin1a2 10455. 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 10447*	Lemma for fin1a2 10455. 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 10448*	Lemma for fin1a2 10455. 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 10449	Lemma for fin1a2 10455. 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 10450*	Lemma for fin1a2 10455. (Contributed by Stefan O'Rear, 8-Nov-2014.)
⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝐴 ∪ {∅}))
Theorem	fin1a2lem12 10451	Lemma for fin1a2 10455. (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 10452	Lemma for fin1a2 10455. (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 10453	Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 10455. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinII)
Theorem	fin1a2s 10454*	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 10455	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 10456*	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 10457*	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 10458*	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 10459*	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 10460*	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 10461*	The union of all union iterates creates the transitive closure; compare trcl 9768. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))    ⇒   ⊢ (TC‘𝐴) = ∪ ran (𝑈‘𝐴)
Theorem	ituniiun 10462*	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 10463*	Lemma for hsmex 10472. 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 10464*	Lemma for hsmex 10472. 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 10465*	Lemma for hsmex 10472. 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 10466	Lemma for hsmex 10472. 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 10467*	Lemma for hsmex 10472. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 10615 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 10468*	Lemma for hsmex 10472. 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 10469*	Lemma for hsmex 10472. 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 10470*	Lemma for hsmex 10472. Combining the above constraints, along with itunitc 10461 and tcrank 9924, 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 10471*	Lemma for hsmex 10472. (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 10472*	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 9632. (Contributed by Stefan O'Rear, 14-Feb-2015.)
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V)
Theorem	hsmex2 10473*	The set of hereditary size-limited sets, assuming ax-reg 9632. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V)
Theorem	hsmex3 10474*	The set of hereditary size-limited sets, assuming ax-reg 9632, 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 10499, as well as weaker forms such as the axiom of countable choice ax-cc 10475 and dependent choice ax-dc 10486. 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 10475*	The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10517, but is weak enough that it can be proven using DC (see axcc 10498). 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 10476*	Lemma for axcc2 10477. (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 10477*	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 10478*	A possibly more useful version of ax-cc 10475 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 10479*	A version of axcc3 10478 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 10480	An ax-cc 10475 equivalent: every set has choice sets of length ω. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ AC ω = V
Theorem	axcc4dom 10481*	Relax the constraint on axcc4 10479 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 10482*	Lemma for domtriom 10483. (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 10483	Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 10354) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)
Theorem	fin41 10484	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 10485	A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 10475. See dominfac 10613 for a version proved from ax-ac 10499. The axiom of Regularity is used for this proof, via inf3lem6 9673, 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 10486*	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 10561. 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 10487	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 10488*	Lemma for axdc2 10489. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹‘𝑥), and show that the "function" described by ax-dc 10486 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 10489*	An apparent strengthening of ax-dc 10486 (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 10490*	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 10491*	Lemma for axdc3 10494. 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 10486 (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 10486 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 10492*	Simple substitution lemma for axdc3 10494. (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 10493*	Lemma for axdc3 10494. 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 10486 (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 10486 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 10489. See axdc3lem2 10491 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 10494*	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 10495*	Lemma for axdc4 10496. (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 10496*	A more general version of axdc3 10494 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 10497*	Lemma for axcc 10498. (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 10498*	Although CC can be proven trivially using ac5 10517, 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 10499*	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 10502 for a more detailed explanation. Theorem ac2 10501 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 10505 is slightly shorter when the biconditional of ax-ac 10499 is expanded into implication and negation. In axac3 10504 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10721 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 10532, ac5 10517, and ac7 10513. The Axiom of Regularity ax-reg 9632 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as Theorem dfac2b 10171. Equivalents to AC are the well-ordering theorem weth 10535 and Zorn's lemma zorn 10547. See ac4 10515 for comments about stronger versions of AC. In order to avoid uses of ax-reg 9632 for derivation of AC equivalents, we provide ax-ac2 10503 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 10503 from ax-ac 10499 is shown by Theorem axac2 10506, and the reverse derivation by axac 10507. Therefore, new proofs should normally use ax-ac2 10503 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 10500*	Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 10499. (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))