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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | isf34lem7 9801* | Lemma for isfin3-4 9804. (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 9802* | Lemma for isfin3-4 9804. (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 9803* | Inference from isfin3-4 9804. (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 9804* | 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 9805 | 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 9806 | Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa)) | ||
Theorem | isfin1-2 9807 | 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 9808 | 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 9809 | 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 9810 | Compact quantifier-free version of the standard definition df-fin 8513. (Contributed by Stefan O'Rear, 6-Jan-2015.) |
⊢ Fin = ( ≈ “ ω) | ||
Theorem | fin23 9811 |
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 9740) 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 9868 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 9812 | Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ (𝐴 ∈ FinIII → 𝐴 ∈ FinIV) | ||
Theorem | isfin5-2 9813 | 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 9814 | 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 9815 | 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 9816 | Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinVII) | ||
Theorem | fin67 9817 | 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 9818 | 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 9819 | 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 9820 | Class form of isfin7-2 9818. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ FinVII = (Fin ∪ (V ∖ dom card)) | ||
Theorem | dfacfin7 9821 | 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 9822 | Lemma for fin1a2 9837. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) ⇒ ⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴) | ||
Theorem | fin1a2lem2 9823 | Lemma for fin1a2 9837. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) ⇒ ⊢ 𝑆:On–1-1→On | ||
Theorem | fin1a2lem3 9824 | Lemma for fin1a2 9837. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) ⇒ ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴)) | ||
Theorem | fin1a2lem4 9825 | Lemma for fin1a2 9837. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) ⇒ ⊢ 𝐸:ω–1-1→ω | ||
Theorem | fin1a2lem5 9826 | Lemma for fin1a2 9837. (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 9827 | Lemma for fin1a2 9837. 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 9828* | Lemma for fin1a2 9837. 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 9829* | Lemma for fin1a2 9837. 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 9830* | Lemma for fin1a2 9837. 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 9831 | Lemma for fin1a2 9837. 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 9832* | Lemma for fin1a2 9837. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝐴 ∪ {∅})) | ||
Theorem | fin1a2lem12 9833 | Lemma for fin1a2 9837. (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 9834 | Lemma for fin1a2 9837. (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 9835 | Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 9837. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinII) | ||
Theorem | fin1a2s 9836* | 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 9837 | 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) | ||
Theorem | itunifval 9838* | 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 9839* | 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 9840* | 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 9841* | 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 9842* | 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 9843* | The union of all union iterates creates the transitive closure; compare trcl 9170. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω)) ⇒ ⊢ (TC‘𝐴) = ∪ ran (𝑈‘𝐴) | ||
Theorem | ituniiun 9844* | 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 9845* | Lemma for hsmex 9854. 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 9846* | Lemma for hsmex 9854. 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 9847* | Lemma for hsmex 9854. 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 9848 | Lemma for hsmex 9854. 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 9849* | Lemma for hsmex 9854. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9997 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 9850* | Lemma for hsmex 9854. 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 9851* | Lemma for hsmex 9854. 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 9852* | Lemma for hsmex 9854. Combining the above constraints, along with itunitc 9843 and tcrank 9313, 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 9853* | Lemma for hsmex 9854. (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 9854* | 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 9056. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V) | ||
Theorem | hsmex2 9855* | The set of hereditary size-limited sets, assuming ax-reg 9056. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V) | ||
Theorem | hsmex3 9856* | The set of hereditary size-limited sets, assuming ax-reg 9056, 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) | ||
In this section we add the Axiom of Choice ax-ac 9881, as well as weaker forms such as the axiom of countable choice ax-cc 9857 and dependent choice ax-dc 9868. 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. | ||
Axiom | ax-cc 9857* | The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 9899, but is weak enough that it can be proven using DC (see axcc 9880). 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 9858* | Lemma for axcc2 9859. (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 9859* | 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 9860* | A possibly more useful version of ax-cc 9857 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 9861* | A version of axcc3 9860 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 9862 | An ax-cc 9857 equivalent: every set has choice sets of length ω. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ AC ω = V | ||
Theorem | axcc4dom 9863* | Relax the constraint on axcc4 9861 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 9864* | Lemma for domtriom 9865. (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 9865 | Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 9736) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω) | ||
Theorem | fin41 9866 | 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 9867 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 9857. See dominfac 9995 for a version proved from ax-ac 9881. The axiom of Regularity is used for this proof, via inf3lem6 9096, 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) | ||
Axiom | ax-dc 9868* | 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 9943. 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 9869 | 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 9870* | Lemma for axdc2 9871. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹‘𝑥), and show that the "function" described by ax-dc 9868 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 9871* | An apparent strengthening of ax-dc 9868 (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 9872* | 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 9873* | Lemma for axdc3 9876. 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 9868 (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 9868 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 9874* | Simple substitution lemma for axdc3 9876. (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 9875* | Lemma for axdc3 9876. 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 9868 (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 9868 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 9871. See axdc3lem2 9873 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 9876* | 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 9877* | Lemma for axdc4 9878. (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 9878* | A more general version of axdc3 9876 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 9879* | Lemma for axcc 9880. (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 9880* | Although CC can be proven trivially using ac5 9899, 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)) | ||
Axiom | ax-ac 9881* |
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 9884 for a more detailed explanation. Theorem ac2 9883 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 9887 is slightly shorter when the biconditional of ax-ac 9881 is expanded into implication and negation. In axac3 9886 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10103 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 9914, ac5 9899, and ac7 9895. The Axiom of Regularity ax-reg 9056 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2b 9556. Equivalents to AC are the well-ordering theorem weth 9917 and Zorn's lemma zorn 9929. See ac4 9897 for comments about stronger versions of AC. In order to avoid uses of ax-reg 9056 for derivation of AC equivalents, we provide ax-ac2 9885 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 9885 from ax-ac 9881 is shown by theorem axac2 9888, and the reverse derivation by axac 9889. Therefore, new proofs should normally use ax-ac2 9885 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 9882* | Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 9881. (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 9883* | 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 9884 is easier to understand.) Note: aceq0 9544 shows the logical equivalence to ax-ac 9881. (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 9884* |
Axiom of Choice using abbreviations. The logical equivalence to ax-ac 9881
can be established by chaining aceq0 9544 and aceq2 9545. A standard
textbook version of AC is derived from this one in dfac2a 9555, and this
version of AC is derived from the textbook version in dfac2b 9556, showing
their logical equivalence (see dfac2 9557).
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 9081. The key theorem for this (used in the proof of dfac2b 9556) is preleq 9079. 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 9885* | In order to avoid uses of ax-reg 9056 for derivation of AC equivalents, we provide ax-ac2 9885, 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 9887. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1796 available. The derivation of ax-ac2 9885 from ax-ac 9881 is shown by theorem axac2 9888, and the reverse derivation by axac 9889. Note that we use ax-reg 9056 to derive ax-ac 9881 from ax-ac2 9885, but not to derive ax-ac2 9885 from ax-ac 9881. (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 9886 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 9885 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.) |
⊢ CHOICE | ||
Theorem | ackm 9887* |
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 9592. 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 9592).
The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html 9592 http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html 9592. (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 9888* | Derive ax-ac2 9885 from ax-ac 9881. (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 9889* | Derive ax-ac 9881 from ax-ac2 9885. Note that ax-reg 9056 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 9890 | Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (CHOICE ↔ ∀𝑥𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | cardeqv 9891 | All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ dom card = V | ||
Theorem | numth3 9892 | All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card) | ||
Theorem | numth2 9893* | 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 9894* | 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 9895* | 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 9896* | 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 9897* |
Equivalent of Axiom of Choice. We do not insist that 𝑓 be a
function. However, theorem ac5 9899, 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 9913. (Contributed by NM, 21-Jul-1996.) |
⊢ ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) | ||
Theorem | ac4c 9898* | Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥) | ||
Theorem | ac5 9899* | 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 9897. (Contributed by NM, 29-Aug-1999.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)) | ||
Theorem | ac5b 9900* | 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)) |
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