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Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiscard3 8901 Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.)
((card‘𝐴) = 𝐴𝐴 ∈ (ω ∪ ran ℵ))

Theoremcardnum 8902 Two ways to express the class of all cardinal numbers, which consists of the finite ordinals in ω plus the transfinite alephs. (Contributed by NM, 10-Sep-2004.)
{𝑥 ∣ (card‘𝑥) = 𝑥} = (ω ∪ ran ℵ)

Theoremalephinit 8903* An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))

Theoremcarduniima 8904 The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
(𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))

Theoremcardinfima 8905* If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
(𝐴𝐵 → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥𝐴 (𝐹𝑥) ∈ ran ℵ) → (𝐹𝐴) ∈ ran ℵ))

Theoremalephiso 8906 Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90. (Contributed by NM, 3-Aug-2004.)
ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})

Theoremalephprc 8907 The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of [TakeutiZaring] p. 90. (Contributed by NM, 11-Nov-2003.)
¬ ran ℵ ∈ V

Theoremalephsson 8908 The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
ran ℵ ⊆ On

Theoremunialeph 8909 The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
ran ℵ = On

Theoremalephsmo 8910 The aleph function is strictly monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
Smo ℵ

Theoremalephf1ALT 8911 Alternate proof of alephf1 8893. (Contributed by Mario Carneiro, 15-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℵ:On–1-1→On

Theoremalephfplem1 8912 Lemma for alephfp 8916. (Contributed by NM, 6-Nov-2004.)
𝐻 = (rec(ℵ, ω) ↾ ω)       (𝐻‘∅) ∈ ran ℵ

Theoremalephfplem2 8913* Lemma for alephfp 8916. (Contributed by NM, 6-Nov-2004.)
𝐻 = (rec(ℵ, ω) ↾ ω)       (𝑤 ∈ ω → (𝐻‘suc 𝑤) = (ℵ‘(𝐻𝑤)))

Theoremalephfplem3 8914* Lemma for alephfp 8916. (Contributed by NM, 6-Nov-2004.)
𝐻 = (rec(ℵ, ω) ↾ ω)       (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)

Theoremalephfplem4 8915 Lemma for alephfp 8916. (Contributed by NM, 5-Nov-2004.)
𝐻 = (rec(ℵ, ω) ↾ ω)        (𝐻 “ ω) ∈ ran ℵ

Theoremalephfp 8916 The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 8917 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)
𝐻 = (rec(ℵ, ω) ↾ ω)       (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω)

Theoremalephfp2 8917 The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 8916 for an actual example of a fixed point. Compare the inequality alephle 8896 that holds in general. Note that if 𝑥 is a fixed point, then ℵ‘ℵ‘ℵ‘... ℵ‘𝑥 = 𝑥. (Contributed by NM, 6-Nov-2004.) (Revised by Mario Carneiro, 15-May-2015.)
𝑥 ∈ On (ℵ‘𝑥) = 𝑥

Theoremalephval3 8918* An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.)
(𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})

Theoremalephsucpw2 8919 The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 9483 or gchaleph2 9479.) The transposed form alephsucpw 9377 cannot be proven without the AC, and is in fact equivalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.)
¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)

Theoremmappwen 8920 Power rule for cardinal arithmetic. Theorem 11.21 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
(((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≈ 𝒫 𝐵)

Theoremfinnisoeu 8921* A finite totally ordered set has a unique order isomorphism to a finite ordinal. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 26-Jun-2015.)
((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))

Theoremiunfictbso 8922 Countability of a countable union of finite sets with a strict (not globally well) order fulfilling the choice role. (Contributed by Stefan O'Rear, 16-Nov-2014.)
((𝐴 ≼ ω ∧ 𝐴 ⊆ Fin ∧ 𝐵 Or 𝐴) → 𝐴 ≼ ω)

2.6.8  Axiom of Choice equivalents

Syntaxwac 8923 Wff for an abbreviation of the axiom of choice.
wff CHOICE

Definitiondf-ac 8924* The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There is a slight problem with taking the exact form of ax-ac 9266 as our definition, because the equivalence to more standard forms (dfac2 8938) requires the Axiom of Regularity, which we often try to avoid. Thus, we take the first of the "textbook forms" as the definition and derive the form of ax-ac 9266 itself as dfac0 8940. (Contributed by Mario Carneiro, 22-Feb-2015.)

(CHOICE ↔ ∀𝑥𝑓(𝑓𝑥𝑓 Fn dom 𝑥))

Theoremaceq1 8925* Equivalence of two versions of the Axiom of Choice ax-ac 9266. The proof uses neither AC nor the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by NM, 5-Apr-2004.)
(∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))

Theoremaceq0 8926* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 9266. (Contributed by NM, 5-Apr-2004.)
(∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)))

Theoremaceq2 8927* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)
(∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))

Theoremaceq3lem 8928* Lemma for dfac3 8929. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝐹 = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢}))       (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))

Theoremdfac3 8929* Equivalence of two versions of the Axiom of Choice. The left-hand side is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Stefan O'Rear, 22-Feb-2015.)
(CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))

Theoremdfac4 8930* Equivalence of two versions of the Axiom of Choice. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
(CHOICE ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))

Theoremdfac5lem1 8931* Lemma for dfac5 8936. (Contributed by NM, 12-Apr-2004.)
(∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))

Theoremdfac5lem2 8932* Lemma for dfac5 8936. (Contributed by NM, 12-Apr-2004.)
𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}       (⟨𝑤, 𝑔⟩ ∈ 𝐴 ↔ (𝑤𝑔𝑤))

Theoremdfac5lem3 8933* Lemma for dfac5 8936. (Contributed by NM, 12-Apr-2004.)
𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}       (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤))

Theoremdfac5lem4 8934* Lemma for dfac5 8936. (Contributed by NM, 11-Apr-2004.)
𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}    &   𝐵 = ( 𝐴𝑦)    &   (𝜑 ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))       (𝜑 → ∃𝑦𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦))

Theoremdfac5lem5 8935* Lemma for dfac5 8936. (Contributed by NM, 12-Apr-2004.)
𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}    &   𝐵 = ( 𝐴𝑦)    &   (𝜑 ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))       (𝜑 → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))

Theoremdfac5 8936* Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))

Theoremdfac2a 8937* Our Axiom of Choice (in the form of ac3 9269) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 8938 for the converse (which does use the Axiom of Regularity). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
(∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → CHOICE)

Theoremdfac2 8938* Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 9269). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 8489 and preleq 8499 that are referenced in the proof each make use of Regularity for their derivations. (The reverse implication can be derived without using Regularity; see dfac2a 8937.) TODO: Fix label in comment, and put label changes into list at top of set.mm. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
(CHOICE ↔ ∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))

Theoremdfac7 8939* Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 9268). The proof does not depend AC on but does depend on the Axiom of Regularity. (Contributed by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))

Theoremdfac0 8940* Equivalence of two versions of the Axiom of Choice. The proof uses the Axiom of Regularity. The right-hand side is our original ax-ac 9266. (Contributed by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)))

Theoremdfac1 8941* Equivalence of two versions of the Axiom of Choice ax-ac 9266. The proof uses the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))

Theoremdfac8 8942* A proof of the equivalency of the Well Ordering Theorem weth 9302 and the Axiom of Choice ac7 9280. (Contributed by Mario Carneiro, 5-Jan-2013.)
(CHOICE ↔ ∀𝑥𝑟 𝑟 We 𝑥)

Theoremdfac9 8943* Equivalence of the axiom of choice with a statement related to ac9 9290; definition AC3 of [Schechter] p. 139. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(CHOICE ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅))

Theoremdfac10 8944 Axiom of Choice equivalent: the cardinality function measures every set. (Contributed by Mario Carneiro, 6-May-2015.)
(CHOICE ↔ dom card = V)

Theoremdfac10c 8945* Axiom of Choice equivalent: every set is equinumerous to an ordinal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
(CHOICE ↔ ∀𝑥𝑦 ∈ On 𝑦𝑥)

Theoremdfac10b 8946 Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 8924). (Contributed by Stefan O'Rear, 17-Jan-2015.)
(CHOICE ↔ ( ≈ “ On) = V)

Theoremacacni 8947 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
((CHOICE𝐴𝑉) → AC 𝐴 = V)

Theoremdfacacn 8948 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
(CHOICE ↔ ∀𝑥AC 𝑥 = V)

Theoremdfac13 8949 The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
(CHOICE ↔ ∀𝑥 𝑥AC 𝑥)

Theoremdfac12lem1 8950* Lemma for dfac12 8956. (Contributed by Mario Carneiro, 29-May-2015.)
(𝜑𝐴 ∈ On)    &   (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)    &   𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))    &   (𝜑𝐶 ∈ On)    &   𝐻 = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶))       (𝜑 → (𝐺𝐶) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))

Theoremdfac12lem2 8951* Lemma for dfac12 8956. (Contributed by Mario Carneiro, 29-May-2015.)
(𝜑𝐴 ∈ On)    &   (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)    &   𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))    &   (𝜑𝐶 ∈ On)    &   𝐻 = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∀𝑧𝐶 (𝐺𝑧):(𝑅1𝑧)–1-1→On)       (𝜑 → (𝐺𝐶):(𝑅1𝐶)–1-1→On)

Theoremdfac12lem3 8952* Lemma for dfac12 8956. (Contributed by Mario Carneiro, 29-May-2015.)
(𝜑𝐴 ∈ On)    &   (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)    &   𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))       (𝜑 → (𝑅1𝐴) ∈ dom card)

Theoremdfac12r 8953 The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 8956 does not assume the Axiom of Regularity. (Contributed by Mario Carneiro, 29-May-2015.)
(∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ (𝑅1 “ On) ⊆ dom card)

Theoremdfac12k 8954* Equivalence of dfac12 8956 and dfac12a 8955, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)
(∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)

Theoremdfac12a 8955 The axiom of choice holds iff every ordinal has a well-orderable powerset. (Contributed by Mario Carneiro, 29-May-2015.)
(CHOICE ↔ ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)

Theoremdfac12 8956 The axiom of choice holds iff every aleph has a well-orderable powerset. (Contributed by Mario Carneiro, 21-May-2015.)
(CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)

Theoremkmlem1 8957* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2. (Contributed by NM, 5-Apr-2004.)
(∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 𝜑) → ∃𝑦𝑧𝑥 𝜓) → ∀𝑥(∀𝑧𝑥𝑤𝑥 𝜑 → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → 𝜓)))

Theoremkmlem2 8958* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
(∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))

Theoremkmlem3 8959* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. The right-hand side is part of the hypothesis of 4. (Contributed by NM, 25-Mar-2004.)
((𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ ↔ ∃𝑣𝑧𝑤𝑥 (𝑧𝑤 → ¬ 𝑣 ∈ (𝑧𝑤)))

Theoremkmlem4 8960* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)

Theoremkmlem5 8961* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)

Theoremkmlem6 8962* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝜑𝐴 = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))

Theoremkmlem7 8963* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)))

Theoremkmlem8 8964* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 4-Apr-2004.)
((¬ ∃𝑧𝑢𝑤𝑧 𝜓 → ∃𝑦𝑧𝑢 (𝑧 ≠ ∅ → ∃!𝑤 𝑤 ∈ (𝑧𝑦))) ↔ (∃𝑧𝑢𝑤𝑧 𝜓 ∨ ∃𝑦𝑦𝑢 ∧ ∀𝑧𝑢 ∃!𝑤 𝑤 ∈ (𝑧𝑦))))

Theoremkmlem9 8965* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)

Theoremkmlem10 8966* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       (∀(∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) → ∃𝑦𝑧𝐴 𝜑)

Theoremkmlem11 8967* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       (𝑧𝑥 → (𝑧 𝐴) = (𝑧 (𝑥 ∖ {𝑧})))

Theoremkmlem12 8968* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       (∀𝑧𝑥 (𝑧 (𝑥 ∖ {𝑧})) ≠ ∅ → (∀𝑧𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 𝐴)))))

Theoremkmlem13 8969* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 5-Apr-2004.)
𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}       (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥(¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))

Theoremkmlem14 8970* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
(𝜑 ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))    &   (𝜓 ↔ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))    &   (𝜒 ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))       (∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑦𝑧𝑣𝑢(𝑦𝑥𝜑))

Theoremkmlem15 8971* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
(𝜑 ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))    &   (𝜓 ↔ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))    &   (𝜒 ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))       ((¬ 𝑦𝑥𝜒) ↔ ∀𝑧𝑣𝑢𝑦𝑥𝜓))

Theoremkmlem16 8972* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
(𝜑 ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))    &   (𝜓 ↔ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))    &   (𝜒 ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))       ((∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥𝜒)) ↔ ∃𝑦𝑧𝑣𝑢((𝑦𝑥𝜑) ∨ (¬ 𝑦𝑥𝜓)))

Theoremdfackm 8973* Equivalence of the Axiom of Choice and Maes' AC ackm 9272. The proof consists of lemmas kmlem1 8957 through kmlem16 8972 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 8936 with biid 251) establishes the AC equivalence shown by Maes' writeup. The left-hand-side AC shown here was chosen because it is shorter to display. (Contributed by NM, 13-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))))

2.6.9  Cardinal number arithmetic

Syntaxccda 8974 Extend class definition to include cardinal number addition.
class +𝑐

Definitiondf-cda 8975* Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 8977 for its value and a description. (Contributed by NM, 24-Sep-2004.)
+𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))

Theoremcdafn 8976 Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)
+𝑐 Fn (V × V)

Theoremcdaval 8977 Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while Cartesian product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 9358, carddom 9361, and cardsdom 9362. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
((𝐴𝑉𝐵𝑊) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))

((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵))

Theoremcdaun 8979 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by NM, 5-Apr-2007.)
((𝐴𝑉𝐵𝑊 ∧ (𝐴𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵))

Theoremcdaen 8980 Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷))

Theoremcdaenun 8981 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴𝐵𝐶𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴 +𝑐 𝐶) ≈ (𝐵𝐷))

Theoremcda1en 8982 Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉 ∧ ¬ 𝐴𝐴) → (𝐴 +𝑐 1𝑜) ≈ suc 𝐴)

Theoremcda1dif 8983 Adding and subtracting one gives back the original set. Similar to pncan 10272 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)

Theorempm110.643 8984 1+1=2 for cardinal number addition, derived from pm54.43 8811 as promised. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 8743), but after applying definitions, our theorem is equivalent. The comment for cdaval 8977 explains why we use instead of =. See pm110.643ALT 8985 for a shorter proof that doesn't use pm54.43 8811. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
(1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Theorempm110.643ALT 8985 Alternate proof of pm110.643 8984. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Theoremcda0en 8986 Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝑉 → (𝐴 +𝑐 ∅) ≈ 𝐴)

Theoremxp2cda 8987 Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝑉 → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴))

Theoremcdacomen 8988 Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)

Theoremcdaassen 8989 Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 +𝑐 𝐵) +𝑐 𝐶) ≈ (𝐴 +𝑐 (𝐵 +𝑐 𝐶)))

Theoremxpcdaen 8990 Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) ≈ ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)))

Theoremmapcdaen 8991 Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))

Theorempwcdaen 8992 Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 +𝑐 𝐵) ≈ (𝒫 𝐴 × 𝒫 𝐵))

Theoremcdadom1 8993 Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝐵 → (𝐴 +𝑐 𝐶) ≼ (𝐵 +𝑐 𝐶))

Theoremcdadom2 8994 Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝐵 → (𝐶 +𝑐 𝐴) ≼ (𝐶 +𝑐 𝐵))

Theoremcdadom3 8995 A set is dominated by its cardinal sum with another. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊) → 𝐴 ≼ (𝐴 +𝑐 𝐵))

Theoremcdaxpdom 8996 Cartesian product dominates disjoint union for sets with cardinality greater than 1. Similar to Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 18-May-2015.)
((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵))

Theoremcdafi 8997 The cardinal sum of two finite sets is finite. (Contributed by NM, 22-Oct-2004.)
((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 +𝑐 𝐵) ≺ ω)

Theoremcdainflem 8998 Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.)
((𝐴𝐵) ≈ ω → (𝐴 ≈ ω ∨ 𝐵 ≈ ω))

Theoremcdainf 8999 A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(ω ≼ 𝐴 ↔ ω ≼ (𝐴 +𝑐 𝐴))

Theoreminfcda1 9000 An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)
(ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
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