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Type | Label | Description |
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Statement | ||
Theorem | hsmex 10301* | 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 9461. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V) | ||
Theorem | hsmex2 10302* | The set of hereditary size-limited sets, assuming ax-reg 9461. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V) | ||
Theorem | hsmex3 10303* | The set of hereditary size-limited sets, assuming ax-reg 9461, 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 10328, as well as weaker forms such as the axiom of countable choice ax-cc 10304 and dependent choice ax-dc 10315. 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 10304* | The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10346, but is weak enough that it can be proven using DC (see axcc 10327). 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 10305* | Lemma for axcc2 10306. (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 10306* | 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 10307* | A possibly more useful version of ax-cc 10304 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 10308* | A version of axcc3 10307 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 10309 | An ax-cc 10304 equivalent: every set has choice sets of length ω. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ AC ω = V | ||
Theorem | axcc4dom 10310* | Relax the constraint on axcc4 10308 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 10311* | Lemma for domtriom 10312. (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 10312 | Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 10183) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω) | ||
Theorem | fin41 10313 | 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 10314 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 10304. See dominfac 10442 for a version proved from ax-ac 10328. The axiom of Regularity is used for this proof, via inf3lem6 9502, 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 10315* | 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 10390. 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 10316 | 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 10317* | Lemma for axdc2 10318. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹‘𝑥), and show that the "function" described by ax-dc 10315 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 10318* | An apparent strengthening of ax-dc 10315 (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 10319* | 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 10320* | Lemma for axdc3 10323. 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 10315 (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 10315 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 10321* | Simple substitution lemma for axdc3 10323. (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 10322* | Lemma for axdc3 10323. 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 10315 (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 10315 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 10318. See axdc3lem2 10320 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 10323* | 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 10324* | Lemma for axdc4 10325. (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 10325* | A more general version of axdc3 10323 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 10326* | Lemma for axcc 10327. (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 10327* | Although CC can be proven trivially using ac5 10346, 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 10328* |
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 10331 for a more detailed explanation. Theorem ac2 10330 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 10334 is slightly shorter when the biconditional of ax-ac 10328 is expanded into implication and negation. In axac3 10333 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10550 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 10361, ac5 10346, and ac7 10342. The Axiom of Regularity ax-reg 9461 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as Theorem dfac2b 9999. Equivalents to AC are the well-ordering theorem weth 10364 and Zorn's lemma zorn 10376. See ac4 10344 for comments about stronger versions of AC. In order to avoid uses of ax-reg 9461 for derivation of AC equivalents, we provide ax-ac2 10332 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 10332 from ax-ac 10328 is shown by Theorem axac2 10335, and the reverse derivation by axac 10336. Therefore, new proofs should normally use ax-ac2 10332 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 10329* | Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 10328. (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 10330* | 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 10331 is easier to understand.) Note: aceq0 9987 shows the logical equivalence to ax-ac 10328. (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 10331* |
Axiom of Choice using abbreviations. The logical equivalence to ax-ac 10328
can be established by chaining aceq0 9987 and aceq2 9988. A standard
textbook version of AC is derived from this one in dfac2a 9998, and this
version of AC is derived from the textbook version in dfac2b 9999, showing
their logical equivalence (see dfac2 10000).
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 9487. The key theorem for this (used in the proof of dfac2b 9999) is preleq 9485. 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 10332* | In order to avoid uses of ax-reg 9461 for derivation of AC equivalents, we provide ax-ac2 10332, 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 10334. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1797 available. The derivation of ax-ac2 10332 from ax-ac 10328 is shown by Theorem axac2 10335, and the reverse derivation by axac 10336. Note that we use ax-reg 9461 to derive ax-ac 10328 from ax-ac2 10332, but not to derive ax-ac2 10332 from ax-ac 10328. (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 10333 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10332 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.) |
⊢ CHOICE | ||
Theorem | ackm 10334* |
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 10035. 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 10035.
The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html 10035 http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html 10035. (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 10335* | Derive ax-ac2 10332 from ax-ac 10328. (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 10336* | Derive ax-ac 10328 from ax-ac2 10332. Note that ax-reg 9461 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 10337 | Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (CHOICE ↔ ∀𝑥𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | cardeqv 10338 | All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ dom card = V | ||
Theorem | numth3 10339 | All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card) | ||
Theorem | numth2 10340* | 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 10341* | 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 10342* | 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 10343* | 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 10344* |
Equivalent of Axiom of Choice. We do not insist that 𝑓 be a
function. However, Theorem ac5 10346, 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 10360. (Contributed by NM, 21-Jul-1996.) |
⊢ ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) | ||
Theorem | ac4c 10345* | Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥) | ||
Theorem | ac5 10346* | 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 10344. (Contributed by NM, 29-Aug-1999.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)) | ||
Theorem | ac5b 10347* | 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 10348* | A version of ac6 10349 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 10349* | 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 10353, 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 10350* | 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 10351* | 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 10352* | 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 10353* | Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9762, we derive this strong version of ac6 10349 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 10354* | Equivalent of Axiom of Choice. Contrapositive of ac6s 10353. (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 10355* | Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 10356. (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 10356* | 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 10357* | ac6s 10353 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 10358* | Version of ac6 10349 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 10359* | 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 10360* | 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 10361* | 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 10362* | 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 9760). (Contributed by NM, 29-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) | ||
Theorem | numthcor 10363* | Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) | ||
Theorem | weth 10364* | 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 10365* | Lemma for zorn2 10375. (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 10366* | Lemma for zorn2 10375. (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 10367* | Lemma for zorn2 10375. (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 10368* | Lemma for zorn2 10375. (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 10369* | Lemma for zorn2 10375. (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 10370* | Lemma for zorn2 10375. (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))) | ||
Theorem | zorn2lem7 10371* | Lemma for zorn2 10375. (Contributed by NM, 6-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} ⇒ ⊢ ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑠((𝑠 ⊆ 𝐴 ∧ 𝑅 Or 𝑠) → ∃𝑎 ∈ 𝐴 ∀𝑟 ∈ 𝑠 (𝑟𝑅𝑎 ∨ 𝑟 = 𝑎))) → ∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎𝑅𝑏) | ||
Theorem | zorn2g 10372* | Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10375 avoids the Axiom of Choice by assuming that 𝐴 is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑤((𝑤 ⊆ 𝐴 ∧ 𝑅 Or 𝑤) → ∃𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝑤 (𝑧𝑅𝑥 ∨ 𝑧 = 𝑥))) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) | ||
Theorem | zorng 10373* | Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 10376 avoids the Axiom of Choice by assuming that 𝐴 is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | zornn0g 10374* | Variant of Zorn's lemma zorng 10373 in which ∅, the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | zorn2 10375* | Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set 𝐴 (with an ordering relation 𝑅) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 10365 through zorn2lem7 10371; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 10371. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝑅 Po 𝐴 ∧ ∀𝑤((𝑤 ⊆ 𝐴 ∧ 𝑅 Or 𝑤) → ∃𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝑤 (𝑧𝑅𝑥 ∨ 𝑧 = 𝑥))) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) | ||
Theorem | zorn 10376* | Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 10375 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | zornn0 10377* | Variant of Zorn's lemma zorn 10376 in which ∅, the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | ttukeylem1 10378* | Lemma for ttukey 10387. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)) | ||
Theorem | ttukeylem2 10379* | Lemma for ttukey 10387. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶)) → 𝐷 ∈ 𝐴) | ||
Theorem | ttukeylem3 10380* | Lemma for ttukey 10387. (Contributed by Mario Carneiro, 11-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) & ⊢ 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴, {(𝐹‘∪ dom 𝑧)}, ∅))))) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ On) → (𝐺‘𝐶) = if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)))) | ||
Theorem | ttukeylem4 10381* | Lemma for ttukey 10387. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) & ⊢ 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴, {(𝐹‘∪ dom 𝑧)}, ∅))))) ⇒ ⊢ (𝜑 → (𝐺‘∅) = 𝐵) | ||
Theorem | ttukeylem5 10382* | Lemma for ttukey 10387. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) & ⊢ 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴, {(𝐹‘∪ dom 𝑧)}, ∅))))) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶 ⊆ 𝐷)) → (𝐺‘𝐶) ⊆ (𝐺‘𝐷)) | ||
Theorem | ttukeylem6 10383* | Lemma for ttukey 10387. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) & ⊢ 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴, {(𝐹‘∪ dom 𝑧)}, ∅))))) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ suc (card‘(∪ 𝐴 ∖ 𝐵))) → (𝐺‘𝐶) ∈ 𝐴) | ||
Theorem | ttukeylem7 10384* | Lemma for ttukey 10387. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) & ⊢ 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴, {(𝐹‘∪ dom 𝑧)}, ∅))))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) | ||
Theorem | ttukey2g 10385* | The Teichmüller-Tukey Lemma ttukey 10387 with a slightly stronger conclusion: we can set up the maximal element of 𝐴 so that it also contains some given 𝐵 ∈ 𝐴 as a subset. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) | ||
Theorem | ttukeyg 10386* | The Teichmüller-Tukey Lemma ttukey 10387 stated with the "choice" as an antecedent (the hypothesis ∪ 𝐴 ∈ dom card says that ∪ 𝐴 is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | ttukey 10387* | The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If 𝐴 is a nonempty collection of finite character, then 𝐴 has a maximal element with respect to inclusion. Here "finite character" means that 𝑥 ∈ 𝐴 iff every finite subset of 𝑥 is in 𝐴. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | axdclem 10388* | Lemma for axdc 10390. (Contributed by Mario Carneiro, 25-Jan-2013.) |
⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω) ⇒ ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘𝐾)𝑥𝑧) → (𝐾 ∈ ω → (𝐹‘𝐾)𝑥(𝐹‘suc 𝐾))) | ||
Theorem | axdclem2 10389* | Lemma for axdc 10390. Using the full Axiom of Choice, we can construct a choice function 𝑔 on 𝒫 dom 𝑥. From this, we can build a sequence 𝐹 starting at any value 𝑠 ∈ dom 𝑥 by repeatedly applying 𝑔 to the set (𝐹‘𝑥) (where 𝑥 is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.) |
⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω) ⇒ ⊢ (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))) | ||
Theorem | axdc 10390* | This theorem derives ax-dc 10315 using ax-ac 10328 and ax-inf 9507. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.) |
⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) | ||
Theorem | fodomg 10391 | An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the axiom of choice ac7g 10343. The axiom of choice is not needed for finite sets, see fodomfi 9202. See also fodomnum 9926. (Contributed by NM, 23-Jul-2004.) (Proof shortened by BJ, 20-May-2024.) |
⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) | ||
Theorem | fodom 10392 | An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴) | ||
Theorem | dmct 10393 | The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω) | ||
Theorem | rnct 10394 | The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → ran 𝐴 ≼ ω) | ||
Theorem | fodomb 10395* | Equivalence of an onto mapping and dominance for a nonempty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.) |
⊢ ((𝐴 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) ↔ (∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴)) | ||
Theorem | wdomac 10396 | When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ (𝑋 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌) | ||
Theorem | brdom3 10397* | Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥)) | ||
Theorem | brdom5 10398* | An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥)) | ||
Theorem | brdom4 10399* | An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥)) | ||
Theorem | brdom7disj 10400* | An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝐴 ∩ 𝐵) = ∅ ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)) |
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