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Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdomtriomlem 9601* Lemma for domtriom 9602. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V    &   𝐵 = {𝑦 ∣ (𝑦𝐴𝑦 ≈ 𝒫 𝑛)}    &   𝐶 = (𝑛 ∈ ω ↦ ((𝑏𝑛) ∖ 𝑘𝑛 (𝑏𝑘)))       𝐴 ∈ Fin → ω ≼ 𝐴)

Theoremdomtriom 9602 Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 9473) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V       (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)

Theoremfin41 9603 Under countable choice, the IV-finite sets (Dedekind-finite) coincide with I-finite (finite in the usual sense) sets. (Contributed by Mario Carneiro, 16-May-2015.)
FinIV = Fin

Theoremdominf 9604 A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 9594. See dominfac 9732 for a version proved from ax-ac 9618. The axiom of Regularity is used for this proof, via inf3lem6 8829, and its use is necessary: otherwise the set 𝐴 = {𝐴} or 𝐴 = {∅, 𝐴} (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ 𝐴 𝐴) → ω ≼ 𝐴)

3.1.2  Introduce the Axiom of Dependent Choice

Axiomax-dc 9605* 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 9680. But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013.)
((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))

Theoremdcomex 9606 The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
ω ∈ V

Theoremaxdc2lem 9607* Lemma for axdc2 9608. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹𝑥), and show that the "function" described by ax-dc 9605 can be restricted so that it is a real function (since the stated properties only show that it is the superset of a function). (Contributed by Mario Carneiro, 25-Jan-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝐴 ∈ V    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}    &   𝐺 = (𝑥 ∈ ω ↦ (𝑥))       ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc2 9608* An apparent strengthening of ax-dc 9605 (but derived from it) which shows that there is a denumerable sequence 𝑔 for any function that maps elements of a set 𝐴 to nonempty subsets of 𝐴 such that 𝑔(𝑥 + 1) ∈ 𝐹(𝑔(𝑥)) for all 𝑥 ∈ ω. The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc3lem 9609* 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).) (Unnecessary distinct variable restrictions were removed by David Abernethy, 18-Mar-2014.) (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario Carneiro, 18-Mar-2014.)
𝐴 ∈ V    &   𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}       𝑆 ∈ V

Theoremaxdc3lem2 9610* Lemma for axdc3 9613. 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 9605 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (𝑛):𝑚𝐴 (for some integer 𝑚). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 9605 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence , we can construct the sequence 𝑔 that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.)
𝐴 ∈ V    &   𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}    &   𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})       (∃(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc3lem3 9611* Simple substitution lemma for axdc3 9613. (Contributed by Mario Carneiro, 27-Jan-2013.)
𝐴 ∈ V    &   𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}    &   𝐵 ∈ V       (𝐵𝑆 ↔ ∃𝑚 ∈ ω (𝐵:suc 𝑚𝐴 ∧ (𝐵‘∅) = 𝐶 ∧ ∀𝑘𝑚 (𝐵‘suc 𝑘) ∈ (𝐹‘(𝐵𝑘))))

Theoremaxdc3lem4 9612* Lemma for axdc3 9613. 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 9605 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (𝑛):𝑚𝐴 (for some integer 𝑚). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 9605 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 9608. See axdc3lem2 9610 for the rest of the proof. (Contributed by Mario Carneiro, 27-Jan-2013.)
𝐴 ∈ V    &   𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}    &   𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})       ((𝐶𝐴𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc3 9613* Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value 𝐶. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. (Contributed by Mario Carneiro, 27-Jan-2013.)
𝐴 ∈ V       ((𝐶𝐴𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))

Theoremaxdc4lem 9614* Lemma for axdc4 9615. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
𝐴 ∈ V    &   𝐺 = (𝑛 ∈ ω, 𝑥𝐴 ↦ ({suc 𝑛} × (𝑛𝐹𝑥)))       ((𝐶𝐴𝐹:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝑘𝐹(𝑔𝑘))))

Theoremaxdc4 9615* A more general version of axdc3 9613 that allows the function 𝐹 to vary with 𝑘. (Contributed by Mario Carneiro, 31-Jan-2013.)
𝐴 ∈ V       ((𝐶𝐴𝐹:(ω × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝑘𝐹(𝑔𝑘))))

Theoremaxcclem 9616* Lemma for axcc 9617. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
𝐴 = (𝑥 ∖ {∅})    &   𝐹 = (𝑛 ∈ ω, 𝑦 𝐴 ↦ (𝑓𝑛))    &   𝐺 = (𝑤𝐴 ↦ (‘suc (𝑓𝑤)))       (𝑥 ≈ ω → ∃𝑔𝑧𝑥 (𝑧 ≠ ∅ → (𝑔𝑧) ∈ 𝑧))

Theoremaxcc 9617* Although CC can be proven trivially using ac5 9636, we prove it here using DC. (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Feb-2013.)
(𝑥 ≈ ω → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))

3.2  ZFC Set Theory - add the Axiom of Choice

3.2.1  Introduce the Axiom of Choice

Axiomax-ac 9618* 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 9621 for a more detailed explanation. Theorem ac2 9620 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 9624 is slightly shorter when the biconditional of ax-ac 9618 is expanded into implication and negation. In axac3 9623 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9840 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 9651, ac5 9636, and ac7 9632. The Axiom of Regularity ax-reg 8788 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2b 9288. Equivalents to AC are the well-ordering theorem weth 9654 and Zorn's lemma zorn 9666. See ac4 9634 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 8788 for derivation of AC equivalents, we provide ax-ac2 9622 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 9622 from ax-ac 9618 is shown by theorem axac2 9625, and the reverse derivation by axac 9626. Therefore, new proofs should normally use ax-ac2 9622 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

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

Theoremzfac 9619* Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 9618. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
𝑥𝑦𝑧((𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Theoremac2 9620* 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 9621 is easier to understand.) Note: aceq0 9276 shows the logical equivalence to ax-ac 9618. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)

Theoremac3 9621* Axiom of Choice using abbreviations. The logical equivalence to ax-ac 9618 can be established by chaining aceq0 9276 and aceq2 9277. A standard textbook version of AC is derived from this one in dfac2a 9287, and this version of AC is derived from the textbook version in dfac2b 9288, showing their logical equivalence (see dfac2 9289).

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 8812. The key theorem for this (used in the proof of dfac2b 9288) is preleq 8810. With this modified definition of ordered pair, it can be seen that 𝑦 is actually a choice function on the members of 𝑥.

For example, suppose 𝑥 = {{1, 2}, {1, 3}, {2, 3, 4}}. Let us try 𝑦 = {{{1, 2}, 1}, {{1, 3}, 1}, {{2, 3, 4}, 2}}. For the member (of 𝑥) 𝑧 = {1, 2}, the only assignment to 𝑤 and 𝑣 that satisfies the axiom is 𝑤 = 1 and 𝑣 = {{1, 2}, 1}, so there is exactly one 𝑤 as required. We verify the other two members of 𝑥 similarly. Thus, 𝑦 satisfies the axiom. Using our modified ordered pair definition, we can say that 𝑦 corresponds to the choice function {⟨{1, 2}, 1⟩, ⟨{1, 3}, 1⟩, ⟨{2, 3, 4}, 2⟩}. Of course other choices for 𝑦 will also satisfy the axiom, for example 𝑦 = {{{1, 2}, 2}, {{1, 3}, 1}, {{2, 3, 4}, 4}}. What AC tells us is that there exists at least one such 𝑦, but it doesn't tell us which one.

(New usage is discouraged.) (Contributed by NM, 19-Jul-1996.)

𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣))

Axiomax-ac2 9622* In order to avoid uses of ax-reg 8788 for derivation of AC equivalents, we provide ax-ac2 9622, 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 9624. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1839 available. The derivation of ax-ac2 9622 from ax-ac 9618 is shown by theorem axac2 9625, and the reverse derivation by axac 9626. Note that we use ax-reg 8788 to derive ax-ac 9618 from ax-ac2 9622, but not to derive ax-ac2 9622 from ax-ac 9618. (Contributed by NM, 19-Dec-2016.)
𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))

Theoremaxac3 9623 This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 9622 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
CHOICE

Theoremackm 9624* 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 9325. 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).

The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)

𝑥𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))

Theoremaxac2 9625* Derive ax-ac2 9622 from ax-ac 9618. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))

Theoremaxac 9626* Derive ax-ac 9618 from ax-ac2 9622. Note that ax-reg 8788 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)
𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))

Theoremaxaci 9627 Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥𝜑)       𝜑

Theoremcardeqv 9628 All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
dom card = V

Theoremnumth3 9629 All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
(𝐴𝑉𝐴 ∈ dom card)

Theoremnumth2 9630* Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.)
𝐴 ∈ V       𝑥 ∈ On 𝑥𝐴

Theoremnumth 9631* Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
𝐴 ∈ V       𝑥 ∈ On ∃𝑓 𝑓:𝑥1-1-onto𝐴

Theoremac7 9632* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.)
𝑓(𝑓𝑥𝑓 Fn dom 𝑥)

Theoremac7g 9633* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)
(𝑅𝐴 → ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅))

Theoremac4 9634* Equivalent of Axiom of Choice. We do not insist that 𝑓 be a function. However, theorem ac5 9636, derived from this one, shows that this form of the axiom does imply that at least one such set 𝑓 whose existence we assert is in fact a function. Axiom of Choice of [TakeutiZaring] p. 83.

Takeuti and Zaring call this "weak choice" in contrast to "strong choice" 𝐹𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧), which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable 𝐹 and thus cannot be expressed in our first-order formalization. However, it has been shown that ZF plus strong choice is a conservative extension of ZF plus weak choice. See Ulrich Felgner, "Comparison of the axioms of local and universal choice", Fundamenta Mathematica, 71, 43-62 (1971).

Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 9650. (Contributed by NM, 21-Jul-1996.)

𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)

Theoremac4c 9635* Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.)
𝐴 ∈ V       𝑓𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)

Theoremac5 9636* 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 9634. (Contributed by NM, 29-Aug-1999.)
𝐴 ∈ V       𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))

Theoremac5b 9637* Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)
𝐴 ∈ V       (∀𝑥𝐴 𝑥 ≠ ∅ → ∃𝑓(𝑓:𝐴 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))

Theoremac6num 9638* A version of ac6 9639 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6 9639* 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 9643, allows 𝐵 to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6c4 9640* Equivalent of Axiom of Choice. 𝐵 is a collection 𝐵(𝑥) of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))

Theoremac6c5 9641* Equivalent of Axiom of Choice. 𝐵 is a collection 𝐵(𝑥) of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝐵)

Theoremac9 9642* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)

Theoremac6s 9643* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9055, we derive this strong version of ac6 9639 that doesn't require 𝐵 to be a set. (Contributed by NM, 4-Feb-2004.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6n 9644* Equivalent of Axiom of Choice. Contrapositive of ac6s 9643. (Contributed by NM, 10-Jun-2007.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑓(𝑓:𝐴𝐵 → ∃𝑥𝐴 𝜓) → ∃𝑥𝐴𝑦𝐵 𝜑)

Theoremac6s2 9645* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 9646. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 𝜓))

Theoremac6s3 9646* Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓𝑥𝐴 𝜓)

Theoremac6sg 9647* ac6s 9643 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))

Theoremac6sf 9648* Version of ac6 9639 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)
𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6s4 9649* 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       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))

Theoremac6s5 9650* 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       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝐵)

Theoremac8 9651* 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.)
((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))

Theoremac9s 9652* 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 9053). (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)

3.2.2  AC equivalents: well-ordering, Zorn's lemma

Theoremnumthcor 9653* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)

Theoremweth 9654* 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.)
(𝐴𝑉 → ∃𝑥 𝑥 We 𝐴)

Theoremzorn2lem1 9655* Lemma for zorn2 9665. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)

Theoremzorn2lem2 9656* Lemma for zorn2 9665. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝑦𝑥 → (𝐹𝑦)𝑅(𝐹𝑥)))

Theoremzorn2lem3 9657* Lemma for zorn2 9665. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))

Theoremzorn2lem4 9658* Lemma for zorn2 9665. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑅 Po 𝐴𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)

Theoremzorn2lem5 9659* Lemma for zorn2 9665. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       (((𝑤 We 𝐴𝑥 ∈ On) ∧ ∀𝑦𝑥 𝐻 ≠ ∅) → (𝐹𝑥) ⊆ 𝐴)

Theoremzorn2lem6 9660* Lemma for zorn2 9665. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       (𝑅 Po 𝐴 → (((𝑤 We 𝐴𝑥 ∈ On) ∧ ∀𝑦𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹𝑥)))

Theoremzorn2lem7 9661* Lemma for zorn2 9665. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑠((𝑠𝐴𝑅 Or 𝑠) → ∃𝑎𝐴𝑟𝑠 (𝑟𝑅𝑎𝑟 = 𝑎))) → ∃𝑎𝐴𝑏𝐴 ¬ 𝑎𝑅𝑏)

Theoremzorn2g 9662* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 9665 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((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)

Theoremzorng 9663* 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 9666 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((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremzornn0g 9664* Variant of Zorn's lemma zorng 9663 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((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremzorn2 9665* 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 9655 through zorn2lem7 9661; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 9661. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐴 ∈ V       ((𝑅 Po 𝐴 ∧ ∀𝑤((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)

Theoremzorn 9666* 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 9665 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
𝐴 ∈ V       (∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremzornn0 9667* Variant of Zorn's lemma zorn 9666 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((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremttukeylem1 9668* Lemma for ttukey 9677. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))       (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))

Theoremttukeylem2 9669* Lemma for ttukey 9677. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))       ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)

Theoremttukeylem3 9670* Lemma for ttukey 9677. (Contributed by Mario Carneiro, 11-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (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)}, ∅)))))       ((𝜑𝐶 ∈ On) → (𝐺𝐶) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))

Theoremttukeylem4 9671* Lemma for ttukey 9677. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (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)

Theoremttukeylem5 9672* Lemma for ttukey 9677. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (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 ∧ 𝐶𝐷)) → (𝐺𝐶) ⊆ (𝐺𝐷))

Theoremttukeylem6 9673* Lemma for ttukey 9677. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (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)}, ∅)))))       ((𝜑𝐶 ∈ suc (card‘( 𝐴𝐵))) → (𝐺𝐶) ∈ 𝐴)

Theoremttukeylem7 9674* Lemma for ttukey 9677. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (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 → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))

Theoremttukey2g 9675* The Teichmüller-Tukey Lemma ttukey 9677 with a slightly stronger conclusion: we can set up the maximal element of 𝐴 so that it also contains some given 𝐵𝐴 as a subset. (Contributed by Mario Carneiro, 15-May-2015.)
(( 𝐴 ∈ dom card ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))

Theoremttukeyg 9676* The Teichmüller-Tukey Lemma ttukey 9677 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(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremttukey 9677* 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 𝑥𝐴 iff every finite subset of 𝑥 is in 𝐴. (Contributed by Mario Carneiro, 15-May-2015.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremaxdclem 9678* Lemma for axdc 9680. (Contributed by Mario Carneiro, 25-Jan-2013.)
𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)       ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹𝐾)𝑥𝑧) → (𝐾 ∈ ω → (𝐹𝐾)𝑥(𝐹‘suc 𝐾)))

Theoremaxdclem2 9679* Lemma for axdc 9680. 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 (𝐹𝑥) (where 𝑥 is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)
𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)       (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))

Theoremaxdc 9680* This theorem derives ax-dc 9605 using ax-ac 9618 and ax-inf 8834. 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.)
((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))

Theoremfodom 9681 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 9633. AC is not needed for finite sets - see fodomfi 8529. See also fodomnum 9215. (Contributed by NM, 23-Jul-2004.)
𝐴 ∈ V       (𝐹:𝐴onto𝐵𝐵𝐴)

Theoremfodomg 9682 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵𝐴))

Theoremdmct 9683 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → dom 𝐴 ≼ ω)

Theoremrnct 9684 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → ran 𝐴 ≼ ω)

Theoremfodomb 9685* 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:𝐴onto𝐵) ↔ (∅ ≺ 𝐵𝐵𝐴))

Theoremwdomac 9686 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* 𝑌𝑋𝑌)

Theorembrdom3 9687* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))

Theorembrdom5 9688* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))

Theorembrdom4 9689* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))

Theorembrdom7disj 9690* 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    &   (𝐴𝐵) = ∅       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))

Theorembrdom6disj 9691* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝐴𝐵) = ∅       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))

Theoremfin71ac 9692 Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.)
FinVII = Fin

Theoremimadomg 9693 An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐵 → (Fun 𝐹 → (𝐹𝐴) ≼ 𝐴))

Theoremfimact 9694 The image by a function of a countable set is countable. (Contributed by Thierry Arnoux, 27-Mar-2018.)
((𝐴 ≼ ω ∧ Fun 𝐹) → (𝐹𝐴) ≼ ω)

Theoremfnrndomg 9695 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)
(𝐴𝐵 → (𝐹 Fn 𝐴 → ran 𝐹𝐴))

Theoremfnct 9696 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)

Theoremmptct 9697* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → (𝑥𝐴𝐵) ≼ ω)

Theoremiunfo 9698* Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)
𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)       (2nd𝑇):𝑇onto 𝑥𝐴 𝐵

Theoremiundom2g 9699* An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)    &   (𝜑 𝑥𝐴 (𝐶𝑚 𝐵) ∈ AC 𝐴)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)       (𝜑𝑇 ≼ (𝐴 × 𝐶))

Theoremiundomg 9700* An upper bound for the cardinality of an indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)    &   (𝜑 𝑥𝐴 (𝐶𝑚 𝐵) ∈ AC 𝐴)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)    &   (𝜑 → (𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵)       (𝜑 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))

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